Coverage for local/lib/python2.7/site-packages/sage/coding/linear

"The constructor requires no arguments.\nIt takes the optional arguments ['maximum_error_weight'].\nSee the documentation of sage.coding.linear_code.LinearCodeSyndromeDecoder for more details."

sage: from sage.coding.information_set_decoder import LinearCodeInformationSetDecoder

sage: cl = LinearCodeInformationSetDecoder

sage: _explain_constructor(cl)

"The constructor requires the arguments ['number_errors'].\nIt takes the optional arguments ['algorithm'].\nIt accepts unspecified arguments as well.\nSee the documentation of sage.coding.information_set_decoder.LinearCodeInformationSetDecoder for more details."

"""

import inspect

if inspect.isclass(cl):

argspec = inspect.getargspec(cl.__init__)

skip = 2 # skip the self and code arguments

else:

# Not a class, assume it's a factory function posing as a class

argspec = inspect.getargspec(cl)

skip = 1 # skip code argument

if argspec.defaults:

args = argspec.args[skip:-len(argspec.defaults)]

kwargs = argspec.args[-len(argspec.defaults):]

opts = "It takes the optional arguments {}.".format(kwargs)

else:

args = argspec.args[skip:]

opts = "It takes no optional arguments."

if args:

reqs = "The constructor requires the arguments {}.".format(args)

else:

reqs = "The constructor requires no arguments."

if argspec.varargs or argspec.keywords:

var = "It accepts unspecified arguments as well.\n"

else:

var = ""

return("{}\n{}\n{}See the documentation of {}.{} for more details."\

.format(reqs, opts, var, cl.__module__, cl.__name__))

class AbstractLinearCode(Module):

"""

Abstract class for linear codes.

This class contains all methods that can be used on Linear Codes

and on Linear Codes families.

So, every Linear Code-related class should inherit from this abstract

class.

To implement a linear code, you need to:

- inherit from AbstractLinearCode

- call AbstractLinearCode ``__init__`` method in the subclass constructor. Example:

``super(SubclassName, self).__init__(base_field, length, "EncoderName", "DecoderName")``.

By doing that, your subclass will have its ``length`` parameter

initialized and will be properly set as a member of the category framework.

You need of course to complete the constructor by adding any additional parameter

needed to describe properly the code defined in the subclass.

- Add the following two lines on the class level::

_registered_encoders = {}

_registered_decoders = {}

- fill the dictionary of its encoders in ``sage.coding.__init__.py`` file. Example:

I want to link the encoder ``MyEncoderClass`` to ``MyNewCodeClass``

under the name ``MyEncoderName``.

All I need to do is to write this line in the ``__init__.py`` file:

``MyNewCodeClass._registered_encoders["NameOfMyEncoder"] = MyEncoderClass`` and all instances of

``MyNewCodeClass`` will be able to use instances of ``MyEncoderClass``.

- fill the dictionary of its decoders in ``sage.coding.__init__`` file. Example:

I want to link the encoder ``MyDecoderClass`` to ``MyNewCodeClass``

under the name ``MyDecoderName``.

All I need to do is to write this line in the ``__init__.py`` file:

``MyNewCodeClass._registered_decoders["NameOfMyDecoder"] = MyDecoderClass`` and all instances of

``MyNewCodeClass`` will be able to use instances of ``MyDecoderClass``.

As AbstractLinearCode is not designed to be implemented, it does not have any representation

methods. You should implement ``_repr_`` and ``_latex_`` methods in the subclass.

.. NOTE::

:class:`AbstractLinearCode` has generic implementations of the comparison methods ``__cmp__``

and ``__eq__`` which use the generator matrix and are quite slow. In subclasses you are

encouraged to override these functions.

.. WARNING::

The default encoder should always have `F^{k}` as message space, with `k` the dimension

of the code and `F` is the base ring of the code.

A lot of methods of the abstract class rely on the knowledge of a generator matrix.

It is thus strongly recommended to set an encoder with a generator matrix implemented

as a default encoder.

"""

_registered_encoders = {}

_registered_decoders = {}

def __init__(self, base_field, length, default_encoder_name, default_decoder_name):

"""

Initializes mandatory parameters that any linear code shares.

This method only exists for inheritance purposes as it initializes

parameters that need to be known by every linear code. The class

:class:`sage.coding.linear_code.AbstractLinearCode` should never be

directly instantiated.

INPUT:

- ``base_field`` -- the base field of ``self``

- ``length`` -- the length of ``self`` (a Python int or a Sage Integer, must be > 0)

- ``default_encoder_name`` -- the name of the default encoder of ``self``

- ``default_decoder_name`` -- the name of the default decoder of ``self``

EXAMPLES:

The following example demonstrates how to subclass `AbstractLinearCode`

for representing a new family of codes. The example family is non-sensical::

sage: class MyCodeFamily(sage.coding.linear_code.AbstractLinearCode):

....: def __init__(self, field, length, dimension, generator_matrix):

....: sage.coding.linear_code.AbstractLinearCode.__init__(self,field, length, "GeneratorMatrix", "Syndrome")

....: self._dimension = dimension

....: self._generator_matrix = generator_matrix

....: def generator_matrix(self):

....: return self._generator_matrix

....: def _repr_(self):

....: return "[%d, %d] dummy code over GF(%s)" % (self.length(), self.dimension(), self.base_field().cardinality())

We now instantiate a member of our newly made code family::

sage: generator_matrix = matrix(GF(17), 5, 10,

....: {(i,i):1 for i in range(5)})

sage: C = MyCodeFamily(GF(17), 10, 5, generator_matrix)

We can check its existence and parameters::

sage: C

[10, 5] dummy code over GF(17)

We can check that it is truly a part of the framework category::

sage: C.parent()

sage: C.category()

Category of facade finite dimensional vector spaces with basis over Finite Field of size 17

And any method that works on linear codes works for our new dummy code::

sage: C.minimum_distance()

sage: C.is_self_orthogonal()

False

sage: print(C.divisor()) #long time

TESTS:

If the length field is neither a Python int nor a Sage Integer, it will

raise a exception::

sage: C = MyCodeFamily(GF(17), 10.0, 5, generator_matrix)

Traceback (most recent call last):

...

ValueError: length must be a Python int or a Sage Integer

If the length of the code is not a non-zero positive integer

(See :trac:`21326`), it will raise an exception::

sage: empty_generator_matrix = Matrix(GF(17),0,1)

sage: C = MyCodeFamily(GF(17), 0, 1, empty_generator_matrix)

Traceback (most recent call last):

...

ValueError: length must be a non-zero positive integer

If the name of the default decoder is not known by the class, it will raise

a exception::

sage: class MyCodeFamily2(sage.coding.linear_code.AbstractLinearCode):

....: def __init__(self, field, length, dimension, generator_matrix):

....: sage.coding.linear_code.AbstractLinearCode.__init__(self,field, length, "GeneratorMatrix", "Fail")

....: self._dimension = dimension

....: self._generator_matrix = generator_matrix

....: def generator_matrix(self):

....: return self._generator_matrix

....: def _repr_(self):

....: return "[%d, %d] dummy code over GF(%s)" % (self.length(), self.dimension(), self.base_field().cardinality())

sage: C = MyCodeFamily2(GF(17), 10, 5, generator_matrix)

Traceback (most recent call last):

...

ValueError: You must set a valid decoder as default decoder for this code, by filling in the dictionary of registered decoders

If the name of the default encoder is not known by the class, it will raise

an exception::

sage: class MyCodeFamily3(sage.coding.linear_code.AbstractLinearCode):

....: def __init__(self, field, length, dimension, generator_matrix):

....: sage.coding.linear_code.AbstractLinearCode.__init__(self,field, length, "Fail", "Syndrome")

....: self._dimension = dimension

....: self._generator_matrix = generator_matrix

....: def generator_matrix(self):

....: return self._generator_matrix

....: def _repr_(self):

....: return "[%d, %d] dummy code over GF(%s)" % (self.length(), self.dimension(), self.base_field().cardinality())

sage: C = MyCodeFamily3(GF(17), 10, 5, generator_matrix)

Traceback (most recent call last):

...

ValueError: You must set a valid encoder as default encoder for this code, by filling in the dictionary of registered encoders

A ring instead of a field::

sage: codes.LinearCode(IntegerModRing(4),matrix.ones(4))

Traceback (most recent call last):

...

ValueError: 'generator' must be defined on a field (not a ring)

"""

### Add here any generic encoder/decoder ###

#This allows any class which inherits from AbstractLinearCode

#to use generic decoders/encoders

self._registered_encoders["Systematic"] = LinearCodeSystematicEncoder

self._registered_decoders["Syndrome"] = LinearCodeSyndromeDecoder

self._registered_decoders["NearestNeighbor"] = LinearCodeNearestNeighborDecoder

from sage.coding.information_set_decoder import LinearCodeInformationSetDecoder

self._registered_decoders["InformationSet"] = LinearCodeInformationSetDecoder

if not isinstance(length, (int, Integer)):

raise ValueError("length must be a Python int or a Sage Integer")

if length <= 0:

raise ValueError("length must be a non-zero positive integer")

if not base_field.is_field():

raise ValueError("'base_field' must be a field (and {} is not one)".format(base_field))

if not default_encoder_name in self._registered_encoders:

raise ValueError("You must set a valid encoder as default encoder for this code, by filling in the dictionary of registered encoders")

if not default_decoder_name in self._registered_decoders:

raise ValueError("You must set a valid decoder as default decoder for this code, by filling in the dictionary of registered decoders")

self._length = Integer(length)

self._default_decoder_name = default_decoder_name

self._default_encoder_name = default_encoder_name

cat = Modules(base_field).FiniteDimensional().WithBasis().Finite()

facade_for = VectorSpace(base_field, self._length)

self.Element = type(facade_for.an_element()) #for when we made this a non-facade parent

Parent.__init__(self, base=base_field, facade=facade_for, category=cat)

def _repr_(self):

r"""

Return an error message requiring to override ``_repr_`` in ``self``.

As one has to implement specific representation methods (`_repr_` and `_latex_`)

when writing a new code class which inherits from :class:`AbstractLinearCode`,

the generic call to `_repr_` has to fail.

EXAMPLES:

This was taken from :trac:`20899` (and thus ensures this method fixes what was

described in this ticket).

We create a new code class, its dedicated encoder

and set appropriate parameters::

sage: from sage.coding.linear_code import AbstractLinearCode

sage: from sage.coding.encoder import Encoder

sage: class MyCode(AbstractLinearCode):

....: _registered_encoders = {}

....: _registered_decoders = {}

....: def __init__(self):

....: super(MyCode, self).__init__(GF(5), 10, "Monkey", "Syndrome")

....: self._dimension = 2

sage: class MonkeyEncoder(Encoder):

....: def __init__(self, C):

....: super(MonkeyEncoder, self).__init__(C)

....: @cached_method

....: def generator_matrix(self):

....: return matrix(GF(5), 2, 10, [ [1]*5 + [0]*5, [0]*5 + [1]*5 ])

sage: MyCode._registered_encoders["Monkey"] = MonkeyEncoder

sage: MyCode._registered_decoders["Syndrome"] = codes.decoders.LinearCodeSyndromeDecoder

We check we get a sensible error message while asking for a string

representation of an instance of our new class:

sage: C = MyCode()

sage: C #random

Traceback (most recent call last):

...

RuntimeError: Please override _repr_ in the implementation of <class '__main__.MyCode_with_category'>

"""

raise RuntimeError("Please override _repr_ in the implementation of {}".format(self.parent()))

def _latex_(self):

r"""

Return an error message requiring to override ``_latex_`` in ``self``.

As one has to implement specific representation methods (`_repr_` and `_latex_`)

when writing a new code class which inherits from :class:`AbstractLinearCode`,

the generic call to `_latex_` has to fail.

EXAMPLES:

This was taken from :trac:`20899` (and thus ensures this method fixes what was

described in this ticket).

We create a new code class, its dedicated encoder

and set appropriate parameters::

sage: from sage.coding.linear_code import AbstractLinearCode

sage: from sage.coding.encoder import Encoder

sage: class MyCode(AbstractLinearCode):

....: _registered_encoders = {}

....: _registered_decoders = {}

....: def __init__(self):

....: super(MyCode, self).__init__(GF(5), 10, "Monkey", "Syndrome")

....: self._dimension = 2

sage: class MonkeyEncoder(Encoder):

....: def __init__(self, C):

....: super(MonkeyEncoder, self).__init__(C)

....: @cached_method

....: def generator_matrix(self):

....: return matrix(GF(5), 2, 10, [ [1]*5 + [0]*5, [0]*5 + [1]*5 ])

sage: MyCode._registered_encoders["Monkey"] = MonkeyEncoder

sage: MyCode._registered_decoders["Syndrome"] = codes.decoders.LinearCodeSyndromeDecoder

We check we get a sensible error message while asking for a string

representation of an instance of our new class:

sage: C = MyCode()

sage: latex(C)

Traceback (most recent call last):

...

RuntimeError: Please override _latex_ in the implementation of <class '__main__.MyCode_with_category'>

"""

raise RuntimeError("Please override _latex_ in the implementation of {}".format(self.parent()))

def _an_element_(self):

r"""

Return an element of the linear code. Currently, it simply returns

the first row of the generator matrix.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.an_element()

(1, 0, 0, 0, 0, 1, 1)

sage: C2 = C.cartesian_product(C)

sage: C2.an_element()

((1, 0, 0, 0, 0, 1, 1), (1, 0, 0, 0, 0, 1, 1))

"""

return self.gens()[0]

def add_decoder(self, name, decoder):

r"""

Adds an decoder to the list of registered decoders of ``self``.

.. NOTE::

This method only adds ``decoder`` to ``self``, and not to any member of the class

of ``self``. To know how to add an :class:`sage.coding.decoder.Decoder`, please refer

to the documentation of :class:`AbstractLinearCode`.

INPUT:

- ``name`` -- the string name for the decoder

- ``decoder`` -- the class name of the decoder

EXAMPLES:

First of all, we create a (very basic) new decoder::

sage: class MyDecoder(sage.coding.decoder.Decoder):

....: def __init__(self, code):

....: super(MyDecoder, self).__init__(code)

....: def _repr_(self):

....: return "MyDecoder decoder with associated code %s" % self.code()

We now create a new code::

sage: C = codes.HammingCode(GF(2), 3)

We can add our new decoder to the list of available decoders of C::

sage: C.add_decoder("MyDecoder", MyDecoder)

sage: sorted(C.decoders_available())

['InformationSet', 'MyDecoder', 'NearestNeighbor', 'Syndrome']

We can verify that any new code will not know MyDecoder::

sage: C2 = codes.HammingCode(GF(2), 3)

sage: sorted(C2.decoders_available())

['InformationSet', 'NearestNeighbor', 'Syndrome']

TESTS:

It is impossible to use a name which is in the dictionary of available decoders::

sage: C.add_decoder("Syndrome", MyDecoder)

Traceback (most recent call last):

...

ValueError: There is already a registered decoder with this name

"""

if self._registered_decoders == self.__class__._registered_decoders:

self._registered_decoders = copy(self._registered_decoders)

reg_dec = self._registered_decoders

if name in reg_dec:

raise ValueError("There is already a registered decoder with this name")

reg_dec[name] = decoder

else:

if name in self._registered_decoders:

raise ValueError("There is already a registered decoder with this name")

reg_dec[name] = decoder

def add_encoder(self, name, encoder):

r"""

Adds an encoder to the list of registered encoders of ``self``.

.. NOTE::

This method only adds ``encoder`` to ``self``, and not to any member of the class

of ``self``. To know how to add an :class:`sage.coding.encoder.Encoder`, please refer

to the documentation of :class:`AbstractLinearCode`.

INPUT:

- ``name`` -- the string name for the encoder

- ``encoder`` -- the class name of the encoder

EXAMPLES:

First of all, we create a (very basic) new encoder::

sage: class MyEncoder(sage.coding.encoder.Encoder):

....: def __init__(self, code):

....: super(MyEncoder, self).__init__(code)

....: def _repr_(self):

....: return "MyEncoder encoder with associated code %s" % self.code()

We now create a new code::

sage: C = codes.HammingCode(GF(2), 3)

We can add our new encoder to the list of available encoders of C::

sage: C.add_encoder("MyEncoder", MyEncoder)

sage: sorted(C.encoders_available())

['MyEncoder', 'ParityCheck', 'Systematic']

We can verify that any new code will not know MyEncoder::

sage: C2 = codes.HammingCode(GF(2), 3)

sage: sorted(C2.encoders_available())

['ParityCheck', 'Systematic']

TESTS:

It is impossible to use a name which is in the dictionary of available encoders::

sage: C.add_encoder("ParityCheck", MyEncoder)

Traceback (most recent call last):

...

ValueError: There is already a registered encoder with this name

"""

if self._registered_encoders == self.__class__._registered_encoders:

self._registered_encoders = copy(self._registered_encoders)

reg_enc = self._registered_encoders

if name in reg_enc:

raise ValueError("There is already a registered encoder with this name")

reg_enc[name] = encoder

else:

if name in self._registered_encoders:

raise ValueError("There is already a registered encoder with this name")

reg_enc[name] = encoder

def automorphism_group_gens(self, equivalence="semilinear"):

r"""

Return generators of the automorphism group of ``self``.

INPUT:

- ``equivalence`` (optional) -- which defines the acting group, either

* ``permutational``

* ``linear``

* ``semilinear``

OUTPUT:

- generators of the automorphism group of ``self``

- the order of the automorphism group of ``self``

EXAMPLES::

sage: C = codes.HammingCode(GF(4, 'z'), 3)

sage: C.automorphism_group_gens()

([((z, 1, z, z, z, z + 1, 1, z + 1, 1, 1, 1, z + 1, 1, z + 1, z + 1, z + 1, 1, z, 1, z + 1, z); (1,9,5,15,20,13,4)(2,8,12,7,10,14,16,3,21,18,19,6,11,17), Ring endomorphism of Finite Field in z of size 2^2

Defn: z |--> z + 1),

((z, z, z, z, z, 1, z + 1, 1, z + 1, z + 1, z + 1, 1, z, z + 1, z, z, 1, z + 1, 1, 1, 1); (1,10,20,16,6,3,11,19,15,8,5,9,17,12,13)(4,7,21,14,18), Ring endomorphism of Finite Field in z of size 2^2

Defn: z |--> z),

((z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z); (), Ring endomorphism of Finite Field in z of size 2^2

Defn: z |--> z)],

362880)

sage: C.automorphism_group_gens(equivalence="linear")

([((z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, 1, z + 1, z + 1, z + 1, z + 1, z, z + 1, z + 1, z + 1, z + 1, 1, z + 1, z + 1, z + 1); (1,3,17,20,12,16)(2,18)(4,11,21,9,14,10)(5,15,19)(6,8), Ring endomorphism of Finite Field in z of size 2^2

Defn: z |--> z),

((z + 1, z, 1, z + 1, z, z + 1, z + 1, z, 1, z + 1, z, z + 1, z, 1, z, z, z + 1, z, 1, 1, z); (1,15,18,20,13,7,21,17,9,11,5,14,19,4,2,16,10,6,8,3,12), Ring endomorphism of Finite Field in z of size 2^2

Defn: z |--> z),

((z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1); (), Ring endomorphism of Finite Field in z of size 2^2

Defn: z |--> z)],

181440)

sage: C.automorphism_group_gens(equivalence="permutational")

([((1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1); (1,11)(3,10)(4,9)(5,7)(12,21)(14,20)(15,19)(16,17), Ring endomorphism of Finite Field in z of size 2^2

Defn: z |--> z), ((1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1); (2,18)(3,19)(4,10)(5,16)(8,13)(9,14)(11,21)(15,20), Ring endomorphism of Finite Field in z of size 2^2

Defn: z |--> z), ((1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1); (1,19)(3,17)(4,21)(5,20)(7,14)(9,12)(10,16)(11,15), Ring endomorphism of Finite Field in z of size 2^2

Defn: z |--> z), ((1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1); (2,13)(3,14)(4,20)(5,11)(8,18)(9,19)(10,15)(16,21), Ring endomorphism of Finite Field in z of size 2^2

Defn: z |--> z)], 64)

"""

aut_group_can_label = self._canonize(equivalence)

return aut_group_can_label.get_autom_gens(), \

aut_group_can_label.get_autom_order()

def ambient_space(self):

r"""

Returns the ambient vector space of `self`.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.ambient_space()

Vector space of dimension 7 over Finite Field of size 2

"""

return VectorSpace(self.base_ring(),self.length())

def assmus_mattson_designs(self, t, mode=None):

r"""

Assmus and Mattson Theorem (section 8.4, page 303 of [HP2003]_): Let

`A_0, A_1, ..., A_n` be the weights of the codewords in a binary

linear `[n , k, d]` code `C`, and let `A_0^*, A_1^*, ..., A_n^*` be

the weights of the codewords in its dual `[n, n-k, d^*]` code `C^*`.

Fix a `t`, `0<t<d`, and let

.. MATH::

s = |\{ i\ |\ A_i^* \not= 0, 0< i \leq n-t\}|.

Assume `s\leq d-t`.

1. If `A_i\not= 0` and `d\leq i\leq n`

then `C_i = \{ c \in C\ |\ wt(c) = i\}` holds a simple t-design.

2. If `A_i^*\not= 0` and `d*\leq i\leq n-t` then

`C_i^* = \{ c \in C^*\ |\ wt(c) = i\}` holds a simple t-design.

A block design is a pair `(X,B)`, where `X` is a non-empty finite set

of `v>0` elements called points, and `B` is a non-empty finite

multiset of size b whose elements are called blocks, such that each

block is a non-empty finite multiset of `k` points. `A` design without

repeated blocks is called a simple block design. If every subset of

points of size `t` is contained in exactly `\lambda` blocks the block

design is called a `t-(v,k,\lambda)` design (or simply a `t`-design

when the parameters are not specified). When `\lambda=1` then the

block design is called a `S(t,k,v)` Steiner system.

In the Assmus and Mattson Theorem (1), `X` is the set `\{1,2,...,n\}`

of coordinate locations and `B = \{supp(c)\ |\ c \in C_i\}` is the set

of supports of the codewords of `C` of weight `i`. Therefore, the

parameters of the `t`-design for `C_i` are

t = given

v = n

k = i (k not to be confused with dim(C))

b = Ai

lambda = b*binomial(k,t)/binomial(v,t) (by Theorem 8.1.6,

p 294, in [HP2003]_)

Setting the ``mode="verbose"`` option prints out the values of the

parameters.

The first example below means that the binary [24,12,8]-code C has

the property that the (support of the) codewords of weight 8 (resp.,

12, 16) form a 5-design. Similarly for its dual code `C^*` (of course

`C=C^*` in this case, so this info is extraneous). The test fails to

produce 6-designs (ie, the hypotheses of the theorem fail to hold,

not that the 6-designs definitely don't exist). The command

assmus_mattson_designs(C,5,mode="verbose") returns the same value

but prints out more detailed information.

The second example below illustrates the blocks of the 5-(24, 8, 1)

design (i.e., the S(5,8,24) Steiner system).

EXAMPLES::

sage: C = codes.GolayCode(GF(2)) # example 1

sage: C.assmus_mattson_designs(5)

['weights from C: ',

[8, 12, 16, 24],

'designs from C: ',

[[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)], [5, (24, 24, 1)]],

'weights from C*: ',

[8, 12, 16],

'designs from C*: ',

[[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)]]]

sage: C.assmus_mattson_designs(6)

sage: X = range(24) # example 2

sage: blocks = [c.support() for c in C if c.hamming_weight()==8]; len(blocks) # long time computation

759

"""

C = self

ans = []

G = C.generator_matrix()

n = len(G.columns())

Cp = C.dual_code()

wts = C.weight_distribution()

d = min([i for i in range(1,len(wts)) if wts[i]!=0])

if t>=d:

return 0

nonzerowts = [i for i in range(len(wts)) if wts[i]!=0 and i<=n and i>=d]

if mode=="verbose":

for w in nonzerowts:

print("The weight w={} codewords of C* form a t-(v,k,lambda) design, where\n \

t={}, v={}, k={}, lambda={}. \nThere are {} block of this design.".format(\

w,t,n,w,wts[w]*binomial(w,t)//binomial(n,t),wts[w]))

wtsp = Cp.weight_distribution()

dp = min([i for i in range(1,len(wtsp)) if wtsp[i]!=0])

nonzerowtsp = [i for i in range(len(wtsp)) if wtsp[i]!=0 and i<=n-t and i>=dp]

s = len([i for i in range(1,n) if wtsp[i]!=0 and i<=n-t and i>0])

if mode=="verbose":

for w in nonzerowtsp:

print("The weight w={} codewords of C* form a t-(v,k,lambda) design, where\n \

t={}, v={}, k={}, lambda={}. \nThere are {} block of this design.".format(\

w,t,n,w,wts[w]*binomial(w,t)//binomial(n,t),wts[w]))

if s<=d-t:

des = [[t,(n,w,wts[w]*binomial(w,t)//binomial(n,t))] for w in nonzerowts]

ans = ans + ["weights from C: ",nonzerowts,"designs from C: ",des]

desp = [[t,(n,w,wtsp[w]*binomial(w,t)//binomial(n,t))] for w in nonzerowtsp]

ans = ans + ["weights from C*: ",nonzerowtsp,"designs from C*: ",desp]

return ans

return 0

def base_field(self):

r"""

Return the base field of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0], [1,0,0,1,1,0,0], [0,1,0,1,0,1,0], [1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: C.base_field()

Finite Field of size 2

"""

return self.base_ring()

def basis(self):

r"""

Returns a basis of `self`.

OUTPUT:

- ``Sequence`` - an immutable sequence whose universe is ambient space of `self`.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.basis()

[

(1, 0, 0, 0, 0, 1, 1),

(0, 1, 0, 0, 1, 0, 1),

(0, 0, 1, 0, 1, 1, 0),

(0, 0, 0, 1, 1, 1, 1)

]

sage: C.basis().universe()

Vector space of dimension 7 over Finite Field of size 2

"""

gens = self.gens()

from sage.structure.sequence import Sequence

return Sequence(gens, universe=self.ambient_space(), check = False, immutable=True, cr=True)

# S. Pancratz, 19 Jan 2010: In the doctests below, I removed the example

# ``C.binomial_moment(3)``, which was also marked as ``#long``. This way,

# we shorten the doctests time while still maintaining a zero and a

# non-zero example.

def binomial_moment(self, i):

r"""

Returns the i-th binomial moment of the `[n,k,d]_q`-code `C`:

.. MATH::

B_i(C) = \sum_{S, |S|=i} \frac{q^{k_S}-1}{q-1}

where `k_S` is the dimension of the shortened code `C_{J-S}`,

`J=[1,2,...,n]`. (The normalized binomial moment is

`b_i(C) = \binom(n,d+i)^{-1}B_{d+i}(C)`.) In other words, `C_{J-S}`

is isomorphic to the subcode of C of codewords supported on S.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.binomial_moment(2)

sage: C.binomial_moment(4) # long time

.. warning::

This is slow.

REFERENCE:

.. [Du04] \I. Duursma, "Combinatorics of the two-variable zeta function",

Finite fields and applications, 109-136, Lecture Notes in

Comput. Sci., 2948, Springer, Berlin, 2004.

"""

n = self.length()

k = self.dimension()

d = self.minimum_distance()

F = self.base_ring()

q = F.order()

J = range(1,n+1)

Cp = self.dual_code()

dp = Cp.minimum_distance()

if i<d:

return 0

if i>n-dp and i<=n:

return binomial(n,i)*(q**(i+k-n) -1)//(q-1)

from sage.combinat.set_partition import SetPartitions

P = SetPartitions(J,2).list()

b = QQ(0)

for p in P:

p = list(p)

S = p[0]

if len(S)==n-i:

C_S = self.shortened(S)

k_S = C_S.dimension()

b = b + (q**(k_S) -1)//(q-1)

return b

@cached_method

def _canonize(self, equivalence):

r"""

Compute a canonical representative and the automorphism group

under the action of the semimonomial transformation group.

INPUT:

- ``equivalence`` -- which defines the acting group, either

* ``permutational``

* ``linear``

* ``semilinear``

EXAMPLES::

sage: C = codes.HammingCode(GF(4, 'z'), 3)

sage: aut_group_can_label = C._canonize("semilinear")

sage: C_iso = LinearCode(aut_group_can_label.get_transporter()*C.generator_matrix())

sage: C_iso == aut_group_can_label.get_canonical_form()

True

sage: aut_group_can_label.get_autom_gens()

[((z, 1, z + 1, z, 1, 1, z, 1, z + 1, 1, 1, z, 1, z + 1, z, 1, z, 1, z + 1, z + 1, 1); (1,10,8,21,3,20,2,4,6,18,14,9,12,16,17)(5,15,13,7,19), Ring endomorphism of Finite Field in z of size 2^2

Defn: z |--> z),

((z + 1, z, z, z + 1, 1, 1, 1, z + 1, z + 1, z, 1, z, z + 1, 1, 1, z + 1, z + 1, 1, 1, z, z); (1,18,17,5,16,3,10,11,8,21,7,12,9,4)(2,19,15,14,6,20,13), Ring endomorphism of Finite Field in z of size 2^2

Defn: z |--> z + 1),

((z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z); (), Ring endomorphism of Finite Field in z of size 2^2

Defn: z |--> z)]

"""

from sage.coding.codecan.autgroup_can_label import LinearCodeAutGroupCanLabel

return LinearCodeAutGroupCanLabel(self, algorithm_type=equivalence)

def canonical_representative(self, equivalence="semilinear"):

r"""

Compute a canonical orbit representative under the action of the

semimonomial transformation group.

See :mod:`sage.coding.codecan.autgroup_can_label`

for more details, for example if you would like to compute

a canonical form under some more restrictive notion of equivalence,

i.e. if you would like to restrict the permutation group

to a Young subgroup.

INPUT:

- ``equivalence`` (optional) -- which defines the acting group, either

* ``permutational``

* ``linear``

* ``semilinear``

OUTPUT:

- a canonical representative of ``self``

- a semimonomial transformation mapping ``self`` onto its representative

EXAMPLES::

sage: F.<z> = GF(4)

sage: C = codes.HammingCode(F, 3)

sage: CanRep, transp = C.canonical_representative()

Check that the transporter element is correct::

sage: LinearCode(transp*C.generator_matrix()) == CanRep

True

Check if an equivalent code has the same canonical representative::

sage: f = F.hom([z**2])

sage: C_iso = LinearCode(C.generator_matrix().apply_map(f))

sage: CanRep_iso, _ = C_iso.canonical_representative()

sage: CanRep_iso == CanRep

True

Since applying the Frobenius automorphism could be extended to an

automorphism of `C`, the following must also yield ``True``::

sage: CanRep1, _ = C.canonical_representative("linear")

sage: CanRep2, _ = C_iso.canonical_representative("linear")

sage: CanRep2 == CanRep1

True

TESTS:

Check that interrupting this does not segfault

(see :trac:`21651`)::

sage: C = LinearCode(random_matrix(GF(47), 25, 35))

sage: alarm(0.5); C.canonical_representative()

Traceback (most recent call last):

...

AlarmInterrupt

"""

aut_group_can_label = self._canonize(equivalence)

return aut_group_can_label.get_canonical_form(), \

aut_group_can_label.get_transporter()

def __contains__(self, v):

r"""

Returns True if `v` can be coerced into `self`. Otherwise, returns False.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: vector((1, 0, 0, 0, 0, 1, 1)) in C # indirect doctest

True

sage: vector((1, 0, 0, 0, 2, 1, 1)) in C # indirect doctest

True

sage: vector((1, 0, 0, 0, 0, 1/2, 1)) in C # indirect doctest

False

"""

if not v in self.ambient_space() or len(v) != self.length():

return False

return self.syndrome(v) == 0

def characteristic(self):

r"""

Returns the characteristic of the base ring of `self`.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.characteristic()

"""

return (self.base_ring()).characteristic()

def characteristic_polynomial(self):

r"""

Returns the characteristic polynomial of a linear code, as defined in

[Lin1999]_.

EXAMPLES::

sage: C = codes.GolayCode(GF(2))

sage: C.characteristic_polynomial()

-4/3*x^3 + 64*x^2 - 2816/3*x + 4096

"""

R = PolynomialRing(QQ,"x")

x = R.gen()

C = self

Cd = C.dual_code()

Sd = Cd.support()

k = C.dimension()

n = C.length()

q = (C.base_ring()).order()

return q**(n-k)*prod([1-x/j for j in Sd if j>0])

def chinen_polynomial(self):

"""

Returns the Chinen zeta polynomial of the code.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.chinen_polynomial() # long time

1/5*(2*sqrt(2)*t^3 + 2*sqrt(2)*t^2 + 2*t^2 + sqrt(2)*t + 2*t + 1)/(sqrt(2) + 1)

sage: C = codes.GolayCode(GF(3), False)

sage: C.chinen_polynomial() # long time

1/7*(3*sqrt(3)*t^3 + 3*sqrt(3)*t^2 + 3*t^2 + sqrt(3)*t + 3*t + 1)/(sqrt(3) + 1)

This last output agrees with the corresponding example given in

Chinen's paper below.

REFERENCES:

- Chinen, K. "An abundance of invariant polynomials satisfying the

Riemann hypothesis", April 2007 preprint.

"""

from sage.functions.all import sqrt

C = self

n = C.length()

RT = PolynomialRing(QQ,2,"Ts")

T,s = RT.fraction_field().gens()

t = PolynomialRing(QQ,"t").gen()

Cd = C.dual_code()

k = C.dimension()

q = (C.base_ring()).characteristic()

d = C.minimum_distance()

dperp = Cd.minimum_distance()

if dperp > d:

P = RT(C.zeta_polynomial())

# Sage does not find dealing with sqrt(int) *as an algebraic object*

# an easy thing to do. Some tricky gymnastics are used to

# make Sage deal with objects over QQ(sqrt(q)) nicely.

if is_even(n):

Pd = q**(k-n//2) * RT(Cd.zeta_polynomial()) * T**(dperp - d)

else:

Pd = s * q**(k-(n+1)//2) * RT(Cd.zeta_polynomial()) * T**(dperp - d)

CP = P+Pd

f = CP/CP(1,s)

return f(t,sqrt(q))

if dperp < d:

P = RT(C.zeta_polynomial())*T**(d - dperp)

if is_even(n):

Pd = q**(k-n/2)*RT(Cd.zeta_polynomial())

if not(is_even(n)):

Pd = s*q**(k-(n+1)/2)*RT(Cd.zeta_polynomial())

CP = P+Pd

f = CP/CP(1,s)

return f(t,sqrt(q))

if dperp == d:

P = RT(C.zeta_polynomial())

if is_even(n):

Pd = q**(k-n/2)*RT(Cd.zeta_polynomial())

if not(is_even(n)):

Pd = s*q**(k-(n+1)/2)*RT(Cd.zeta_polynomial())

CP = P+Pd

f = CP/CP(1,s)

return f(t,sqrt(q))

@cached_method

def parity_check_matrix(self):

r"""

Returns the parity check matrix of ``self``.

The parity check matrix of a linear code `C` corresponds to the

generator matrix of the dual code of `C`.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: Cperp = C.dual_code()

sage: C; Cperp

[7, 4] Hamming Code over GF(2)

[7, 3] linear code over GF(2)

sage: C.generator_matrix()

[1 0 0 0 0 1 1]

[0 1 0 0 1 0 1]

[0 0 1 0 1 1 0]

[0 0 0 1 1 1 1]

sage: C.parity_check_matrix()

[1 0 1 0 1 0 1]

[0 1 1 0 0 1 1]

[0 0 0 1 1 1 1]

sage: Cperp.parity_check_matrix()

[1 0 0 0 0 1 1]

[0 1 0 0 1 0 1]

[0 0 1 0 1 1 0]

[0 0 0 1 1 1 1]

sage: Cperp.generator_matrix()

[1 0 1 0 1 0 1]

[0 1 1 0 0 1 1]

[0 0 0 1 1 1 1]

"""

G = self.generator_matrix()

H = G.right_kernel()

M = H.basis_matrix()

M.set_immutable()

return M

@cached_method

def covering_radius(self):

r"""

Return the minimal integer `r` such that any element in the ambient space of ``self`` has distance at most `r` to a codeword of ``self``.

This method requires the optional GAP package Guava.

If the covering radius a code equals its minimum distance, then the code is called perfect.

.. NOTE::

This method is currently not implemented on codes over base fields

of cardinality greater than 256 due to limitations in the underlying

algorithm of GAP.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 5)

sage: C.covering_radius() # optional - gap_packages (Guava package)

sage: C = codes.random_linear_code(GF(263), 5, 1)

sage: C.covering_radius() # optional - gap_packages (Guava package)

Traceback (most recent call last):

...

NotImplementedError: the GAP algorithm that Sage is using is limited to computing with fields of size at most 256

"""

if not is_package_installed('gap_packages'):

raise PackageNotFoundError('gap_packages')

gap.load_package("guava")

F = self.base_ring()

if F.cardinality() > 256:

raise NotImplementedError("the GAP algorithm that Sage is using "

"is limited to computing with fields "

"of size at most 256")

gapG = gap(self.generator_matrix())

C = gapG.GeneratorMatCode(gap(F))

r = C.CoveringRadius()

try:

return ZZ(r)

except TypeError:

raise RuntimeError("the covering radius of this code cannot be computed by Guava")

def decode_to_code(self, word, decoder_name=None, *args, **kwargs):

r"""

Corrects the errors in ``word`` and returns a codeword.

INPUT:

- ``word`` -- a vector of the same length as ``self`` over

the base field of ``self``

- ``decoder_name`` -- (default: ``None``) Name of the decoder which will be used

to decode ``word``. The default decoder of ``self`` will be used if

default value is kept.

- ``args``, ``kwargs`` -- all additional arguments are forwarded to :meth:`decoder`

OUTPUT:

- A vector of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: word = vector(GF(2), (1, 1, 0, 0, 1, 1, 0))

sage: w_err = word + vector(GF(2), (1, 0, 0, 0, 0, 0, 0))

sage: C.decode_to_code(w_err)

(1, 1, 0, 0, 1, 1, 0)

It is possible to manually choose the decoder amongst the list of the available ones::

sage: sorted(C.decoders_available())

['InformationSet', 'NearestNeighbor', 'Syndrome']

sage: C.decode_to_code(w_err, 'NearestNeighbor')

(1, 1, 0, 0, 1, 1, 0)

"""

D = self.decoder(decoder_name, *args, **kwargs)

return D.decode_to_code(word)

def decode_to_message(self, word, decoder_name=None, *args, **kwargs):

r"""

Correct the errors in word and decodes it to the message space.

INPUT:

- ``word`` -- a vector of the same length as ``self`` over the

base field of ``self``

- ``decoder_name`` -- (default: ``None``) Name of the decoder which will be used

to decode ``word``. The default decoder of ``self`` will be used if

default value is kept.

- ``args``, ``kwargs`` -- all additional arguments are forwarded to :meth:`decoder`

OUTPUT:

- A vector of the message space of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: word = vector(GF(2), (1, 1, 0, 0, 1, 1, 0))

sage: C.decode_to_message(word)

(0, 1, 1, 0)

It is possible to manually choose the decoder amongst the list of the available ones::

sage: sorted(C.decoders_available())

['InformationSet', 'NearestNeighbor', 'Syndrome']

sage: C.decode_to_message(word, 'NearestNeighbor')

(0, 1, 1, 0)

"""

return self.unencode(self.decode_to_code(word, decoder_name, *args, **kwargs), **kwargs)

@cached_method

def decoder(self, decoder_name=None, *args, **kwargs):

r"""

Return a decoder of ``self``.

INPUT:

- ``decoder_name`` -- (default: ``None``) name of the decoder which will be

returned. The default decoder of ``self`` will be used if

default value is kept.

- ``args``, ``kwargs`` -- all additional arguments will be forwarded to the constructor of the decoder

that will be returned by this method

OUTPUT:

- a decoder object

Besides creating the decoder and returning it, this method also stores

the decoder in a cache. With this behaviour, each decoder will be created

at most one time for ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: C.decoder()

Syndrome decoder for [7, 4] linear code over GF(2) handling errors of weight up to 1

If the name of a decoder which is not known by ``self`` is passed,

an exception will be raised::

sage: sorted(C.decoders_available())

['InformationSet', 'NearestNeighbor', 'Syndrome']

sage: C.decoder('Try')

Traceback (most recent call last):

...

ValueError: There is no Decoder named 'Try'. The known Decoders are: ['InformationSet', 'NearestNeighbor', 'Syndrome']

Some decoders take extra arguments. If the user forgets to supply these,

the error message attempts to be helpful::

sage: C.decoder('InformationSet')

Traceback (most recent call last):

...

ValueError: Constructing the InformationSet decoder failed, possibly due to missing or incorrect parameters.

The constructor requires the arguments ['number_errors'].

It takes the optional arguments ['algorithm'].

It accepts unspecified arguments as well.

See the documentation of sage.coding.information_set_decoder.LinearCodeInformationSetDecoder for more details.

"""

if decoder_name is None:

decoder_name = self._default_decoder_name

if decoder_name in self._registered_decoders:

decClass = self._registered_decoders[decoder_name]

try:

return decClass(self, *args, **kwargs)

except TypeError:

raise ValueError(

"Constructing the {0} decoder failed, possibly due "

"to missing or incorrect parameters.\n{1}".format(

decoder_name, _explain_constructor(decClass)))

else:

raise ValueError(

"There is no Decoder named '{0}'. The known Decoders are: "

"{1}".format(decoder_name, self.decoders_available()))

def decoders_available(self, classes=False):

r"""

Returns a list of the available decoders' names for ``self``.

INPUT:

- ``classes`` -- (default: ``False``) if ``classes`` is set to ``True``,

return instead a ``dict`` mapping available decoder name to the

associated decoder class.

OUTPUT: a list of strings, or a `dict` mapping strings to classes.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: C.decoders_available()

['InformationSet', 'NearestNeighbor', 'Syndrome']

sage: dictionary = C.decoders_available(True)

sage: sorted(dictionary.keys())

['InformationSet', 'NearestNeighbor', 'Syndrome']

sage: dictionary['NearestNeighbor']

"""

if classes:

return copy(self._registered_decoders)

return sorted(self._registered_decoders)

def divisor(self):

r"""

Returns the greatest common divisor of the weights of the nonzero codewords.

EXAMPLES::

sage: C = codes.GolayCode(GF(2))

sage: C.divisor() # Type II self-dual

sage: C = codes.QuadraticResidueCodeEvenPair(17,GF(2))[0]

sage: C.divisor()

"""

C = self

A = C.weight_distribution()

n = C.length()

V = VectorSpace(QQ,n+1)

S = V(A).nonzero_positions()

S0 = [S[i] for i in range(1,len(S))]

if len(S)>1: return GCD(S0)

return 1

def is_projective(self):

r"""

Test whether the code is projective.

A linear code `C` over a field is called *projective* when its dual `Cd`

has minimum weight `\geq 3`, i.e. when no two coordinate positions of

`C` are linearly independent (cf. definition 3 from [BS2011]_ or 9.8.1 from

[BH12]_).

EXAMPLES::

sage: C = codes.GolayCode(GF(2), False)

sage: C.is_projective()

True

sage: C.dual_code().minimum_distance()

A non-projective code::

sage: C = codes.LinearCode(matrix(GF(2),[[1,0,1],[1,1,1]]))

sage: C.is_projective()

False

"""

M = self.generator_matrix().transpose()

R = self.base_field()

def projectivize(row):

if not row.is_zero():

for i in range(len(row)):

if row[i]:

break

row = ~(row[i]) * row

row.set_immutable()

return row

rows = set()

for row in M.rows():

row = projectivize(row)

if row in rows:

return False

rows.add(row)

return True

def dual_code(self):

r"""

Returns the dual code `C^{\perp}` of the code `C`,

.. MATH::

C^{\perp} = \{ v \in V\ |\ v\cdot c = 0,\ \forall c \in C \}.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.dual_code()

[7, 3] linear code over GF(2)

sage: C = codes.HammingCode(GF(4, 'a'), 3)

sage: C.dual_code()

[21, 3] linear code over GF(4)

"""

return LinearCode(self.parity_check_matrix())

def dimension(self):

r"""

Returns the dimension of this code.

EXAMPLES::

sage: G = matrix(GF(2),[[1,0,0],[1,1,0]])

sage: C = LinearCode(G)

sage: C.dimension()

TESTS:

Check that :trac:`21156` is fixed::

sage: from sage.coding.linear_code import AbstractLinearCode

sage: from sage.coding.encoder import Encoder

sage: class MonkeyCode(AbstractLinearCode):

....: _registered_encoders = {}

....: _registered_decoders = {}

....: def __init__(self):

....: super(MonkeyCode, self).__init__(GF(5), 10, "Monkey", "Syndrome")

....:

sage: class MonkeyEncoder(Encoder):

....: def __init__(self, code):

....: super(MonkeyEncoder, self).__init__(C)

....: @cached_method

....: def generator_matrix(self):

....: G = identity_matrix(GF(5), 5).augment(matrix(GF(5), 5, 7))

....: return G

....:

sage: MonkeyCode._registered_encoders["Monkey"] = MonkeyEncoder

sage: C = MonkeyCode()

sage: C.dimension()

"""

try:

return self._dimension

except AttributeError:

dimension = self.generator_matrix().nrows()

self._dimension = dimension

return self._dimension

def direct_sum(self, other):

"""

Returns the code given by the direct sum of the codes ``self`` and

``other``, which must be linear codes defined over the same base ring.

EXAMPLES::

sage: C1 = codes.HammingCode(GF(2), 3)

sage: C2 = C1.direct_sum(C1); C2

[14, 8] linear code over GF(2)

sage: C3 = C1.direct_sum(C2); C3

[21, 12] linear code over GF(2)

"""

C1 = self; C2 = other

G1 = C1.generator_matrix()

G2 = C2.generator_matrix()

F = C1.base_ring()

n1 = len(G1.columns())

k1 = len(G1.rows())

n2 = len(G2.columns())

k2 = len(G2.rows())

MS1 = MatrixSpace(F,k2,n1)

MS2 = MatrixSpace(F,k1,n2)

Z1 = MS1(0)

Z2 = MS2(0)

top = G1.augment(Z2)

bottom = Z1.augment(G2)

G = top.stack(bottom)

return LinearCode(G)

def __eq__(self, right):

"""

Checks if ``self`` is equal to ``right``.

EXAMPLES::

sage: C1 = codes.HammingCode(GF(2), 3)

sage: C2 = codes.HammingCode(GF(2), 3)

sage: C1 == C2

True

TESTS:

We check that :trac:`16644` is fixed::

sage: C = codes.HammingCode(GF(2), 3)

sage: C == ZZ

False

"""

if not (isinstance(right, LinearCode)\

and self.length() == right.length()\

and self.dimension() == right.dimension()\

and self.base_ring() == right.base_ring()):

return False

Ks = self.parity_check_matrix().right_kernel()

rbas = right.gens()

if not all(c in Ks for c in rbas):

return False

Kr = right.parity_check_matrix().right_kernel()

sbas = self.gens()

if not all(c in Kr for c in sbas):

return False

return True

def __ne__(self, other):

r"""

Tests inequality of ``self`` and ``other``.

This is a generic implementation, which returns the inverse of ``__eq__`` for self.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C1 = LinearCode(G)

sage: C2 = LinearCode(G)

sage: C1 != C2

False

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,1,1]])

sage: C2 = LinearCode(G)

sage: C1 != C2

True

"""

return not self == other

def encode(self, word, encoder_name=None, *args, **kwargs):

r"""

Transforms an element of a message space into a codeword.

INPUT:

- ``word`` -- a vector of a message space of the code.

- ``encoder_name`` -- (default: ``None``) Name of the encoder which will be used

to encode ``word``. The default encoder of ``self`` will be used if

default value is kept.

- ``args``, ``kwargs`` -- all additional arguments are forwarded to the construction of the

encoder that is used.

.. NOTE::

The default encoder always has `F^{k}` as message space, with `k` the dimension

of ``self`` and `F` the base ring of ``self``.

One can use the following shortcut to encode a word ::

C(word)

OUTPUT:

- a vector of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: word = vector((0, 1, 1, 0))

sage: C.encode(word)

(1, 1, 0, 0, 1, 1, 0)

sage: C(word)

(1, 1, 0, 0, 1, 1, 0)

It is possible to manually choose the encoder amongst the list of the available ones::

sage: sorted(C.encoders_available())

['GeneratorMatrix', 'Systematic']

sage: word = vector((0, 1, 1, 0))

sage: C.encode(word, 'GeneratorMatrix')

(1, 1, 0, 0, 1, 1, 0)

"""

E = self.encoder(encoder_name, *args, **kwargs)

return E.encode(word)

def __call__(self, m):

r"""

Returns either ``m`` if it is a codeword or ``self.encode(m)``

if it is an element of the message space of the encoder used by

``encode``.

INPUT:

- ``m`` -- a vector whose length equals to code's length or an element

of the message space used by ``encode``

- ``**kwargs`` -- extra arguments are forwarded to ``encode``

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: word = vector((0, 1, 1, 0))

sage: C(word)

(1, 1, 0, 0, 1, 1, 0)

sage: c = C.random_element()

sage: C(c) == c

True

TESTS:

If one passes a vector which belongs to the ambient space, it has to be a codeword.

Otherwise, an exception is raised::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: word = vector((0, 1, 1, 0, 0, 1, 0))

sage: C(word)

Traceback (most recent call last):

...

ValueError: If the input is a vector which belongs to the ambient space, it has to be a codeword

"""

if m in self.ambient_space():

if m in self:

return m

else:

raise ValueError("If the input is a vector which belongs to the ambient space, it has to be a codeword")

else:

return self.encode(m)

@cached_method

def encoder(self, encoder_name=None, *args, **kwargs):

r"""

Returns an encoder of ``self``.

The returned encoder provided by this method is cached.

This methods creates a new instance of the encoder subclass designated by ``encoder_name``.

While it is also possible to do the same by directly calling the subclass' constructor,

it is strongly advised to use this method to take advantage of the caching mechanism.

INPUT:

- ``encoder_name`` -- (default: ``None``) name of the encoder which will be

returned. The default encoder of ``self`` will be used if

default value is kept.

- ``args``, ``kwargs`` -- all additional arguments are forwarded to the constructor of the encoder

this method will return.

OUTPUT:

- an Encoder object.

.. NOTE::

The default encoder always has `F^{k}` as message space, with `k` the dimension

of ``self`` and `F` the base ring of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: C.encoder()

Generator matrix-based encoder for [7, 4] linear code over GF(2)

We check that the returned encoder is cached::

sage: C.encoder.is_in_cache()

True

If the name of an encoder which is not known by ``self`` is passed,

an exception will be raised::

sage: sorted(C.encoders_available())

['GeneratorMatrix', 'Systematic']

sage: C.encoder('NonExistingEncoder')

Traceback (most recent call last):

...

ValueError: There is no Encoder named 'NonExistingEncoder'. The known Encoders are: ['GeneratorMatrix', 'Systematic']

Some encoders take extra arguments. If the user incorrectly supplies

these, the error message attempts to be helpful::

sage: C.encoder('Systematic', strange_parameter=True)

Traceback (most recent call last):

...

ValueError: Constructing the Systematic encoder failed, possibly due to missing or incorrect parameters.

The constructor requires no arguments.

It takes the optional arguments ['systematic_positions'].

See the documentation of sage.coding.linear_code.LinearCodeSystematicEncoder for more details.

"""

if encoder_name is None:

encoder_name = self._default_encoder_name

if encoder_name in self._registered_encoders:

encClass = self._registered_encoders[encoder_name]

try:

return encClass(self, *args, **kwargs)

except TypeError:

raise ValueError(

"Constructing the {0} encoder failed, possibly due "

"to missing or incorrect parameters.\n{1}".format(

encoder_name, _explain_constructor(encClass)))

else:

raise ValueError(

"There is no Encoder named '{0}'. The known Encoders are: "

"{1}".format(encoder_name, self.encoders_available()))

def encoders_available(self, classes=False):

r"""

Returns a list of the available encoders' names for ``self``.

INPUT:

- ``classes`` -- (default: ``False``) if ``classes`` is set to ``True``,

return instead a ``dict`` mapping available encoder name to the

associated encoder class.

OUTPUT: a list of strings, or a `dict` mapping strings to classes.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: C.encoders_available()

['GeneratorMatrix', 'Systematic']

sage: dictionary = C.encoders_available(True)

sage: sorted(dictionary.items())

[('GeneratorMatrix', <class 'sage.coding.linear_code.LinearCodeGeneratorMatrixEncoder'>),

('Systematic', <class 'sage.coding.linear_code.LinearCodeSystematicEncoder'>)]

"""

if classes:

return copy(self._registered_encoders)

return sorted(self._registered_encoders)

def extended_code(self):

r"""

Returns `self` as an extended code.

See documentation of :class:`sage.coding.extended_code.ExtendedCode`

for details.

EXAMPLES::

sage: C = codes.HammingCode(GF(4,'a'), 3)

sage: C

[21, 18] Hamming Code over GF(4)

sage: Cx = C.extended_code()

sage: Cx

Extension of [21, 18] Hamming Code over GF(4)

"""

from .extended_code import ExtendedCode

return ExtendedCode(self)

def galois_closure(self, F0):

r"""

If ``self`` is a linear code defined over `F` and `F_0` is a subfield

with Galois group `G = Gal(F/F_0)` then this returns the `G`-module

`C^-` containing `C`.

EXAMPLES::

sage: C = codes.HammingCode(GF(4,'a'), 3)

sage: Cc = C.galois_closure(GF(2))

sage: C; Cc

[21, 18] Hamming Code over GF(4)

[21, 20] linear code over GF(4)

sage: c = C.basis()[2]

sage: V = VectorSpace(GF(4,'a'),21)

sage: c2 = V([x^2 for x in c.list()])

sage: c2 in C

False

sage: c2 in Cc

True

"""

G = self.generator_matrix()

F = self.base_ring()

q = F.order()

q0 = F0.order()

a = q.log(q0) # test if F/F0 is a field extension

if not isinstance(a, Integer):

raise ValueError("Base field must be an extension of given field %s"%F0)

n = len(G.columns())

k = len(G.rows())

G0 = [[x**q0 for x in g.list()] for g in G.rows()]

G1 = [[x for x in g.list()] for g in G.rows()]

G2 = G0+G1

MS = MatrixSpace(F,2*k,n)

G3 = MS(G2)

r = G3.rank()

MS = MatrixSpace(F,r,n)

Grref = G3.echelon_form()

G = MS([Grref.row(i) for i in range(r)])

return LinearCode(G)

def __getitem__(self, i):

r"""

Returns the `i`-th codeword of this code.

The implementation of this depends on the implementation of the

:meth:`.__iter__` method.

The implementation is as follows. Suppose that:

- the primitive element of the base_ring of this code is `a`,

- the prime subfield is `p`,

- the field has order `p^m`,

- the code has dimension `k`,

- and the generator matrix is `G`.

Then the :meth:`.__iter__` method returns the elements in this order:

1. first, the following ordered list is returned:

``[i*a^0 * G[0] for i in range(p)]``

2. Next, the following ordered list is returned:

``[i*a^0 * G[0] + a^1*G[0] for i in range(p)]``

3. This continues till we get

``[(i*a^0 +(p-1)*a^1 +...+ (p-1)*a^(m-1))*G[0] for i in range(p)]``

4. Then, we move to G[1]:

``[i*a^0 * G[0] + a^0*G[1] for i in range(p)]``,

and so on.

Hence the `i`-th element can be obtained by the p-adic expansion

of `i` as ``[i_0, i_1, ...,i_{m-1}, i_m, i_{m+1}, ..., i_{km-1}].``

The element that is generated is:

.. MATH::

\begin{aligned}

& (i_0 a^0 + i_1 a^1 + \cdots + i_{m-1} a^{m-1}) G[0] + \\

& (i_m a^0 + i_{m+1} a^1 + \cdots + i_{2m-1} a^{m-1}) G[1] + \\

& \vdots\\

& (i_{(k-1)m} a^0 + \cdots + i_{km-1} a^{m-1}) G[k-1]

\end{aligned}

EXAMPLES::

sage: G = Matrix(GF(3), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: C[24]

(2, 2, 0, 1, 2, 2, 0)

sage: C[24] == C.list()[24]

True

TESTS::

sage: C = random_matrix(GF(25,'a'), 2, 7).row_space()

sage: C = LinearCode(C.basis_matrix())

sage: Clist = C.list()

sage: all([C[i]==Clist[i] for i in range(len(C))])

True

Check that only the indices less than the size of the code are

allowed::

sage: C[25**2]

Traceback (most recent call last):

...

IndexError: The value of the index 'i' (=625) must be between

0 and 'q^k -1' (=624), inclusive, where 'q' is the size of the

base field and 'k' is the dimension of the code.

Check that codewords are immutable. See :trac:`16338`::

sage: C[0].is_immutable()

True

"""

# IMPORTANT: If the __iter__() function implementation is changed

# then the implementation here must also be changed so that

# list(self)[i] and self[i] both return the same element.

F = self.base_ring()

maxindex = F.order()**self.dimension()-1

if i < 0 or i > maxindex:

raise IndexError("The value of the index 'i' (={}) must be between "

"0 and 'q^k -1' (={}), inclusive, where 'q' is "

"the size of the base field and 'k' is the "

"dimension of the code.".format(i, maxindex))

a = F.primitive_element()

m = F.degree()

p = F.prime_subfield().order()

A = [a ** k for k in range(m)]

G = self.generator_matrix()

N = self.dimension()*F.degree() # the total length of p-adic vector

ivec = Integer(i).digits(p, padto=N)

codeword = 0

row = 0

for g in G:

codeword += sum(ivec[j+row*m]*A[j] for j in range(m)) * g

row += 1

# The codewords for a specific code can not change. So, we set them

# to be immutable.

codeword.set_immutable()

return codeword

def generator_matrix(self, encoder_name=None, **kwargs):

r"""

Returns a generator matrix of ``self``.

INPUT:

- ``encoder_name`` -- (default: ``None``) name of the encoder which will be

used to compute the generator matrix. The default encoder of ``self``

will be used if default value is kept.

- ``kwargs`` -- all additional arguments are forwarded to the construction of the

encoder that is used.

EXAMPLES::

sage: G = matrix(GF(3),2,[1,-1,1,-1,1,1])

sage: code = LinearCode(G)

sage: code.generator_matrix()

[1 2 1]

[2 1 1]

"""

E = self.encoder(encoder_name, **kwargs)

return E.generator_matrix()

def systematic_generator_matrix(self, systematic_positions=None):

"""

Return a systematic generator matrix of the code.

A generator matrix of a code is called systematic if it contains

a set of columns forming an identity matrix.

INPUT:

- ``systematic_positions`` -- (default: ``None``) if supplied, the set

of systematic positions in the systematic generator matrix. See the

documentation for :class:`LinearCodeSystematicEncoder` details.

EXAMPLES::

sage: G = matrix(GF(3), [[ 1, 2, 1, 0],\

[ 2, 1, 1, 1]])

sage: C = LinearCode(G)

sage: C.generator_matrix()

[1 2 1 0]

[2 1 1 1]

sage: C.systematic_generator_matrix()

[1 2 0 1]

[0 0 1 2]

Specific systematic positions can also be requested:

sage: C.systematic_generator_matrix(systematic_positions=[3,2])

[1 2 0 1]

[1 2 1 0]

"""

systematic_positions = tuple(systematic_positions) if systematic_positions else None

return self.encoder("Systematic", systematic_positions=systematic_positions).generator_matrix()

generator_matrix_systematic = deprecated_function_alias(20835,

systematic_generator_matrix)

@cached_method

def gens(self):

r"""

Returns the generators of this code as a list of vectors.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.gens()

[(1, 0, 0, 0, 0, 1, 1), (0, 1, 0, 0, 1, 0, 1), (0, 0, 1, 0, 1, 1, 0), (0, 0, 0, 1, 1, 1, 1)]

"""

return self.generator_matrix().rows()

def genus(self):

r"""

Returns the "Duursma genus" of the code, `\gamma_C = n+1-k-d`.

EXAMPLES::

sage: C1 = codes.HammingCode(GF(2), 3); C1

[7, 4] Hamming Code over GF(2)

sage: C1.genus()

sage: C2 = codes.HammingCode(GF(4,"a"), 2); C2

[5, 3] Hamming Code over GF(4)

sage: C2.genus()

Since all Hamming codes have minimum distance 3, these computations

agree with the definition, `n+1-k-d`.

"""

d = self.minimum_distance()

n = self.length()

k = self.dimension()

gammaC = n+1-k-d

return gammaC

def __iter__(self):

"""

Return an iterator over the elements of this linear code.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: [list(c) for c in C if c.hamming_weight() < 4]

[[0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1, 1],

[0, 1, 0, 0, 1, 0, 1], [0, 0, 1, 0, 1, 1, 0],

[1, 1, 1, 0, 0, 0, 0], [1, 0, 0, 1, 1, 0, 0],

[0, 1, 0, 1, 0, 1, 0], [0, 0, 1, 1, 0, 0, 1]]

TESTS::

sage: C = codes.HammingCode(GF(2), 3)

sage: L = list(C)

sage: L[10].is_immutable()

True

"""

from sage.modules.finite_submodule_iter import \

FiniteFieldsubspace_iterator

return FiniteFieldsubspace_iterator(self.generator_matrix(), immutable=True)

@cached_method

def information_set(self):

"""

Return an information set of the code.

Return value of this method is cached.

A set of column positions of a generator matrix of a code

is called an information set if the corresponding columns

form a square matrix of full rank.

OUTPUT:

- Information set of a systematic generator matrix of the code.

EXAMPLES::

sage: G = matrix(GF(3),2,[1,2,0,\

2,1,1])

sage: code = LinearCode(G)

sage: code.systematic_generator_matrix()

[1 2 0]

[0 0 1]

sage: code.information_set()

(0, 2)

"""

return self.encoder("Systematic").systematic_positions()

def is_information_set(self, positions):

"""

Return whether the given positions form an information set.

INPUT:

- A list of positions, i.e. integers in the range 0 to `n-1` where `n`

is the length of `self`.

OUTPUT:

- A boolean indicating whether the positions form an information set.

EXAMPLES::

sage: G = matrix(GF(3),2,[1,2,0,\

2,1,1])

sage: code = LinearCode(G)

sage: code.is_information_set([0,1])

False

sage: code.is_information_set([0,2])

True

"""

try:

self.encoder("Systematic", systematic_positions=tuple(positions))

return True

except ValueError:

return False

def is_permutation_automorphism(self,g):

r"""

Returns `1` if `g` is an element of `S_n` (`n` = length of self) and

if `g` is an automorphism of self.

EXAMPLES::

sage: C = codes.HammingCode(GF(3), 3)

sage: g = SymmetricGroup(13).random_element()

sage: C.is_permutation_automorphism(g)

sage: MS = MatrixSpace(GF(2),4,8)

sage: G = MS([[1,0,0,0,1,1,1,0],[0,1,1,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0]])

sage: C = LinearCode(G)

sage: S8 = SymmetricGroup(8)

sage: g = S8("(2,3)")

sage: C.is_permutation_automorphism(g)

sage: g = S8("(1,2,3,4)")

sage: C.is_permutation_automorphism(g)

"""

basis = self.generator_matrix().rows()

H = self.parity_check_matrix()

V = H.column_space()

HGm = H*g.matrix()

for c in basis:

if HGm*c != V(0):

return False

return True

def is_permutation_equivalent(self,other,algorithm=None):

"""

Returns ``True`` if ``self`` and ``other`` are permutation equivalent

codes and ``False`` otherwise.

The ``algorithm="verbose"`` option also returns a permutation (if

``True``) sending ``self`` to ``other``.

Uses Robert Miller's double coset partition refinement work.

EXAMPLES::

sage: P.<x> = PolynomialRing(GF(2),"x")

sage: g = x^3+x+1

sage: C1 = codes.CyclicCode(length = 7, generator_pol = g); C1

[7, 4] Cyclic Code over GF(2)

sage: C2 = codes.HammingCode(GF(2), 3); C2

[7, 4] Hamming Code over GF(2)

sage: C1.is_permutation_equivalent(C2)

True

sage: C1.is_permutation_equivalent(C2,algorithm="verbose")

(True, (3,4)(5,7,6))

sage: C1 = codes.random_linear_code(GF(2), 10, 5)

sage: C2 = codes.random_linear_code(GF(3), 10, 5)

sage: C1.is_permutation_equivalent(C2)

False

"""

from sage.groups.perm_gps.partn_ref.refinement_binary import NonlinearBinaryCodeStruct

F = self.base_ring()

F_o = other.base_ring()

q = F.order()

G = self.generator_matrix()

n = self.length()

n_o = other.length()

if F != F_o or n != n_o:

return False

k = len(G.rows())

MS = MatrixSpace(F,q**k,n)

CW1 = MS(self.list())

CW2 = MS(other.list())

B1 = NonlinearBinaryCodeStruct(CW1)

B2 = NonlinearBinaryCodeStruct(CW2)

ans = B1.is_isomorphic(B2)

if ans is not False:

if algorithm=="verbose":

Sn = SymmetricGroup(n)

return True, Sn([i+1 for i in ans])**(-1)

return True

return False

def is_self_dual(self):

"""

Returns ``True`` if the code is self-dual (in the usual Hamming inner

product) and ``False`` otherwise.

EXAMPLES::

sage: C = codes.GolayCode(GF(2))

sage: C.is_self_dual()

True

sage: C = codes.HammingCode(GF(2), 3)

sage: C.is_self_dual()

False

"""

return self == self.dual_code()

def is_self_orthogonal(self):

"""

Returns ``True`` if this code is self-orthogonal and ``False``

otherwise.

A code is self-orthogonal if it is a subcode of its dual.

EXAMPLES::

sage: C = codes.GolayCode(GF(2))

sage: C.is_self_orthogonal()

True

sage: C = codes.HammingCode(GF(2), 3)

sage: C.is_self_orthogonal()

False

sage: C = codes.QuasiQuadraticResidueCode(11) # optional - gap_packages (Guava package)

sage: C.is_self_orthogonal() # optional - gap_packages (Guava package)

True

"""

return self.is_subcode(self.dual_code())

def is_galois_closed(self):

r"""

Checks if ``self`` is equal to its Galois closure.

EXAMPLES::

sage: C = codes.HammingCode(GF(4,"a"), 3)

sage: C.is_galois_closed()

False

"""

F = self.base_ring()

p = F.characteristic()

return self == self.galois_closure(GF(p))

def is_subcode(self, other):

"""

Returns ``True`` if ``self`` is a subcode of ``other``.

EXAMPLES::

sage: C1 = codes.HammingCode(GF(2), 3)

sage: G1 = C1.generator_matrix()

sage: G2 = G1.matrix_from_rows([0,1,2])

sage: C2 = LinearCode(G2)

sage: C2.is_subcode(C1)

True

sage: C1.is_subcode(C2)

False

sage: C3 = C1.extended_code()

sage: C1.is_subcode(C3)

False

sage: C4 = C1.punctured([1])

sage: C4.is_subcode(C1)

False

sage: C5 = C1.shortened([1])

sage: C5.is_subcode(C1)

False

sage: C1 = codes.HammingCode(GF(9,"z"), 3)

sage: G1 = C1.generator_matrix()

sage: G2 = G1.matrix_from_rows([0,1,2])

sage: C2 = LinearCode(G2)

sage: C2.is_subcode(C1)

True

"""

G = self.generator_matrix()

for r in G.rows():

if not(r in other):

return False

return True

def cardinality(self):

r"""

Return the size of this code.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.cardinality()

sage: len(C)

"""

return self.base_ring().order()**self.dimension()

__len__ = cardinality

def length(self):

r"""

Returns the length of this code.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.length()

"""

return self._length

def list(self):

r"""

Return a list of all elements of this linear code.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: Clist = C.list()

sage: Clist[5]; Clist[5] in C

(1, 0, 1, 0, 1, 0, 1)

True

"""

return [x for x in self]

def _magma_init_(self, magma):

r"""

Retun a string representation in Magma of this linear code.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: Cm = magma(C) # optional - magma, indirect doctest

sage: Cm.MinimumWeight() # optional - magma

"""

G = magma(self.generator_matrix())._ref()

s = 'LinearCode(%s)' % G

return s

@cached_method

def minimum_distance(self, algorithm=None):

r"""

Returns the minimum distance of ``self``.

.. NOTE::

When using GAP, this raises a ``NotImplementedError`` if

the base field of the code has size greater than 256 due

to limitations in GAP.

INPUT:

- ``algorithm`` -- (default: ``None``) the name of the algorithm to use

to perform minimum distance computation. If set to ``None``,

GAP methods will be used. ``algorithm`` can be:

- ``"Guava"``, which will use optional GAP package Guava

OUTPUT:

- Integer, minimum distance of this code

EXAMPLES::

sage: MS = MatrixSpace(GF(3),4,7)

sage: G = MS([[1,1,1,0,0,0,0], [1,0,0,1,1,0,0], [0,1,0,1,0,1,0], [1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: C.minimum_distance()

If ``algorithm`` is provided, then the minimum distance will be

recomputed even if there is a stored value from a previous run.::

sage: C.minimum_distance(algorithm="gap")

sage: C.minimum_distance(algorithm="guava") # optional - gap_packages (Guava package)

TESTS::

sage: C = codes.random_linear_code(GF(4,"a"), 5, 2)

sage: C.minimum_distance(algorithm='something')

Traceback (most recent call last):

...

ValueError: The algorithm argument must be one of None, 'gap' or 'guava'; got 'something'

The field must be size at most 256::

sage: C = codes.random_linear_code(GF(257,"a"), 5, 2)

sage: C.minimum_distance()

Traceback (most recent call last):

...

NotImplementedError: the GAP algorithm that Sage is using

is limited to computing with fields of size at most 256

"""

if algorithm == "guava" and not is_package_installed('gap_packages'):

raise PackageNotFoundError('gap_packages')

# If the minimum distance has already been computed or provided by

# the user then simply return the stored value.

# This is done only if algorithm is None.

if algorithm not in (None, "gap", "guava"):

raise ValueError("The algorithm argument must be one of None, "

"'gap' or 'guava'; got '{0}'".format(algorithm))

F = self.base_ring()

q = F.order()

if q > 256:

raise NotImplementedError("the GAP algorithm that Sage is using "

"is limited to computing with fields "

"of size at most 256")

G = self.generator_matrix()

n = self.length()

k = self.dimension()

if (q == 2 or q == 3) and algorithm=="guava":

gap.load_package("guava")

C = gap(G).GeneratorMatCode(gap(F))

d = C.MinimumWeight()

return ZZ(d)

return self._minimum_weight_codeword(algorithm).hamming_weight()

def _minimum_weight_codeword(self, algorithm = None):

r"""

Returns a minimum weight codeword of ``self``.

INPUT:

- ``algorithm`` -- (default: ``None``) the name of the algorithm to use

to perform minimum weight codeword search. If set to ``None``,

a search using GAP methods will be done. ``algorithm`` can be:

- ``"Guava"``, which will use optional GAP package Guava

REMARKS:

- The code in the default case allows one (for free) to also compute the

message vector `m` such that `m\*G = v`, and the (minimum) distance, as

a triple. however, this output is not implemented.

- The binary case can presumably be done much faster using Robert Miller's

code (see the docstring for the spectrum method). This is also not (yet)

implemented.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0], [1,0,0,1,1,0,0], [0,1,0,1,0,1,0], [1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: C._minimum_weight_codeword()

(0, 1, 0, 1, 0, 1, 0)

TESTS:

We check that :trac:`18480` is fixed::

sage: codes.HammingCode(GF(2), 2).minimum_distance()

AUTHORS:

- David Joyner (11-2005)

"""

G = self.generator_matrix()

n, k = self.length(), self.dimension()

F = self.base_field()

Gmat = self.generator_matrix()._gap_init_()

current_randstate().set_seed_gap()

if algorithm=="guava":

if not is_package_installed('gap_packages'):

raise PackageNotFoundError('gap_packages')

gap.load_package("guava")

from sage.interfaces.gap import gfq_gap_to_sage

gap.eval("G:="+Gmat)

C = gap(Gmat).GeneratorMatCode(F)

cg = C.MinimumDistanceCodeword()

c = [gfq_gap_to_sage(cg[j],F) for j in range(1,n+1)]

return vector(F, c)

q = F.order()

ans = None

dist_min = n + 1

gap.eval('Gmat:='+Gmat)

gap.eval('K:=GF({})'.format(q))

gap.eval('v:=Z({})*{}'.format(q,[0]*n))

for i in range(1,k+1):

gap.eval("P:=AClosestVectorCombinationsMatFFEVecFFECoords(Gmat,K,v,{},1)".format(i))

gap.eval("d:=WeightVecFFE(P[1])")

v = gap("P[1]")

dist = gap("d")

if dist and dist < dist_min:

dist_min = dist

ans = list(v)

if ans is None:

raise RuntimeError("Computation failed due to some GAP error")

# return the result as a vector (and not a 1xn matrix)

return vector(F, ans)

def module_composition_factors(self, gp):

r"""

Prints the GAP record of the Meataxe composition factors module in

Meataxe notation. This uses GAP but not Guava.

EXAMPLES::

sage: MS = MatrixSpace(GF(2),4,8)

sage: G = MS([[1,0,0,0,1,1,1,0],[0,1,1,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0]])

sage: C = LinearCode(G)

sage: gp = C.permutation_automorphism_group()

Now type "C.module_composition_factors(gp)" to get the record printed.

"""

F = self.base_ring()

q = F.order()

gens = gp.gens()

G = self.generator_matrix()

n = len(G.columns())

MS = MatrixSpace(F,n,n)

mats = [] # initializing list of mats by which the gens act on self

Fn = VectorSpace(F, n)

W = Fn.subspace_with_basis(G.rows()) # this is self

for g in gens:

p = MS(g.matrix())

m = [W.coordinate_vector(r*p) for r in G.rows()]

mats.append(m)

mats_str = str(gap([[list(r) for r in m] for m in mats]))

gap.eval("M:=GModuleByMats("+mats_str+", GF("+str(q)+"))")

print(gap("MTX.CompositionFactors( M )"))

def permutation_automorphism_group(self, algorithm="partition"):

r"""

If `C` is an `[n,k,d]` code over `F`, this function computes the

subgroup `Aut(C) \subset S_n` of all permutation automorphisms of `C`.

The binary case always uses the (default) partition refinement

algorithm of Robert Miller.

Note that if the base ring of `C` is `GF(2)` then this is the full

automorphism group. Otherwise, you could use

:meth:`~sage.coding.linear_code.LinearCode.automorphism_group_gens`

to compute generators of the full automorphism group.

INPUT:

- ``algorithm`` - If ``"gap"`` then GAP's MatrixAutomorphism function

(written by Thomas Breuer) is used. The implementation combines an

idea of mine with an improvement suggested by Cary Huffman. If

``"gap+verbose"`` then code-theoretic data is printed out at

several stages of the computation. If ``"partition"`` then the

(default) partition refinement algorithm of Robert Miller is used.

Finally, if ``"codecan"`` then the partition refinement algorithm

of Thomas Feulner is used, which also computes a canonical

representative of ``self`` (call

:meth:`~sage.coding.linear_code.LinearCode.canonical_representative`

to access it).

OUTPUT:

- Permutation automorphism group

EXAMPLES::

sage: MS = MatrixSpace(GF(2),4,8)

sage: G = MS([[1,0,0,0,1,1,1,0],[0,1,1,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0]])

sage: C = LinearCode(G)

sage: C

[8, 4] linear code over GF(2)

sage: G = C.permutation_automorphism_group()

sage: G.order()

144

sage: GG = C.permutation_automorphism_group("codecan")

sage: GG == G

True

A less easy example involves showing that the permutation

automorphism group of the extended ternary Golay code is the

Mathieu group `M_{11}`.

sage: C = codes.GolayCode(GF(3))

sage: M11 = MathieuGroup(11)

sage: M11.order()

7920

sage: G = C.permutation_automorphism_group() # long time (6s on sage.math, 2011)

sage: G.is_isomorphic(M11) # long time

True

sage: GG = C.permutation_automorphism_group("codecan") # long time

sage: GG == G # long time

True

Other examples::

sage: C = codes.GolayCode(GF(2))

sage: G = C.permutation_automorphism_group()

sage: G.order()

244823040

sage: C = codes.HammingCode(GF(2), 5)

sage: G = C.permutation_automorphism_group()

sage: G.order()

9999360

sage: C = codes.HammingCode(GF(3), 2); C

[4, 2] Hamming Code over GF(3)

sage: C.permutation_automorphism_group(algorithm="partition")

Permutation Group with generators [(1,3,4)]

sage: C = codes.HammingCode(GF(4,"z"), 2); C

[5, 3] Hamming Code over GF(4)

sage: G = C.permutation_automorphism_group(algorithm="partition"); G

Permutation Group with generators [(1,3)(4,5), (1,4)(3,5)]

sage: GG = C.permutation_automorphism_group(algorithm="codecan") # long time

sage: GG == G # long time

True

sage: C.permutation_automorphism_group(algorithm="gap") # optional - gap_packages (Guava package)

Permutation Group with generators [(1,3)(4,5), (1,4)(3,5)]

sage: C = codes.GolayCode(GF(3), True)

sage: C.permutation_automorphism_group(algorithm="gap") # optional - gap_packages (Guava package)

Permutation Group with generators [(5,7)(6,11)(8,9)(10,12), (4,6,11)(5,8,12)(7,10,9), (3,4)(6,8)(9,11)(10,12), (2,3)(6,11)(8,12)(9,10), (1,2)(5,10)(7,12)(8,9)]

However, the option ``algorithm="gap+verbose"``, will print out::

Minimum distance: 5 Weight distribution: [1, 0, 0, 0, 0, 132, 132,

0, 330, 110, 0, 24]

Using the 132 codewords of weight 5 Supergroup size: 39916800

in addition to the output of

``C.permutation_automorphism_group(algorithm="gap")``.

"""

F = self.base_ring()

q = F.order()

G = self.generator_matrix() if 2*self.dimension() <= self.length() else self.dual_code().generator_matrix()

n = len(G.columns())

k = len(G.rows())

if "gap" in algorithm:

if not is_package_installed('gap_packages'):

raise PackageNotFoundError('gap_packages')

gap.load_package('guava')

wts = self.weight_distribution() # bottleneck 1

nonzerowts = [i for i in range(len(wts)) if wts[i]!=0]

Sn = SymmetricGroup(n)

Gp = gap("SymmetricGroup(%s)"%n) # initializing G in gap

Gstr = str(gap(G))

gap.eval("C:=GeneratorMatCode("+Gstr+",GF("+str(q)+"))")

gap.eval("eltsC:=Elements(C)")

if algorithm=="gap+verbose":

print("\n Minimum distance: %s \n Weight distribution: \n %s" % (nonzerowts[1], wts))

stop = 0 # only stop if all gens are autos

for i in range(1,len(nonzerowts)):

if stop == 1:

break

wt = nonzerowts[i]

if algorithm=="gap+verbose":

size = Gp.Size()

print("\n Using the %s codewords of weight %s \n Supergroup size: \n %s\n " % (wts[wt], wt, size))

gap.eval("Cwt:=Filtered(eltsC,c->WeightCodeword(c)=%s)"%wt) # bottleneck 2 (repeated

gap.eval("matCwt:=List(Cwt,c->VectorCodeword(c))") # for each i until stop = 1)

if gap("Length(matCwt)") > 0:

A = gap("MatrixAutomorphisms(matCwt)")

G2 = gap("Intersection2(%s,%s)"%(str(A).replace("\n",""),str(Gp).replace("\n",""))) # bottleneck 3

Gp = G2

if Gp.Size()==1:

return PermutationGroup([()])

autgp_gens = Gp.GeneratorsOfGroup()

gens = [Sn(str(x).replace("\n","")) for x in autgp_gens]

stop = 1 # get ready to stop

for x in gens: # if one of these gens is not an auto then don't stop

if not(self.is_permutation_automorphism(x)):

stop = 0

break

G = PermutationGroup(gens)

return G

if algorithm=="partition":

if q == 2:

from sage.groups.perm_gps.partn_ref.refinement_binary import LinearBinaryCodeStruct

B = LinearBinaryCodeStruct(G)

autgp = B.automorphism_group()

L = [[j+1 for j in gen] for gen in autgp[0]]

AutGp = PermutationGroup(L)

else:

from sage.groups.perm_gps.partn_ref.refinement_matrices import MatrixStruct

from sage.matrix.constructor import matrix

weights = {}

for c in self:

wt = c.hamming_weight()

if wt not in weights:

weights[wt] = [c]

else:

weights[wt].append(c)

weights.pop(0)

AutGps = []

for wt, words in iteritems(weights):

M = MatrixStruct(matrix(words))

autgp = M.automorphism_group()

L = [[j+1 for j in gen] for gen in autgp[0]]

G = PermutationGroup(L)

AutGps.append(G)

if len(AutGps) > 0:

AutGp = AutGps[0]

for G in AutGps[1:]:

AutGp = AutGp.intersection(G)

else:

return PermutationGroup([])

return AutGp

if algorithm=="codecan":

gens, _ = self.automorphism_group_gens("permutational")

return PermutationGroup([x.get_perm() for x in gens])

raise NotImplementedError("The only algorithms implemented currently are 'gap', 'gap+verbose', and 'partition'.")

def permuted_code(self, p):

r"""

Returns the permuted code, which is equivalent to ``self`` via the

column permutation ``p``.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: G = C.permutation_automorphism_group(); G

Permutation Group with generators [(4,5)(6,7), (4,6)(5,7), (2,3)(6,7), (2,4)(3,5), (1,2)(5,6)]

sage: g = G("(2,3)(6,7)")

sage: Cg = C.permuted_code(g)

sage: Cg

[7, 4] linear code over GF(2)

sage: C.generator_matrix() == Cg.systematic_generator_matrix()

True

"""

G = copy(self.generator_matrix())

G.permute_columns(p)

return LinearCode(G)

def punctured(self, L):

r"""

Returns a :class:`sage.coding.punctured_code` object from ``L``.

INPUT:

- ``L`` - List of positions to puncture

OUTPUT:

- an instance of :class:`sage.coding.punctured_code`

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.punctured([1,2])

Puncturing of [7, 4] Hamming Code over GF(2) on position(s) [1, 2]

"""

from .punctured_code import PuncturedCode

return PuncturedCode(self, set(L))

def _punctured_form(self, points):

r"""

Returns a representation of self as a :class:`LinearCode` punctured in ``points``.

INPUT:

- ``points`` -- a set of positions where to puncture ``self``

EXAMPLES::

sage: C = codes.random_linear_code(GF(7), 11, 4)

sage: C._punctured_form({3})

[10, 4] linear code over GF(7)

"""

if not isinstance(points, (Integer, int, set)):

raise TypeError("points must be either a Sage Integer, a Python int, or a set")

M = self.generator_matrix()

G = M.delete_columns(list(points))

G = G.echelon_form()

k = G.rank()

return LinearCode(G[:k])

def random_element(self, *args, **kwds):

"""

Returns a random codeword; passes other positional and keyword

arguments to ``random_element()`` method of vector space.

OUTPUT:

- Random element of the vector space of this code

EXAMPLES::

sage: C = codes.HammingCode(GF(4,'a'), 3)

sage: C.random_element() # random test

(1, 0, 0, a + 1, 1, a, a, a + 1, a + 1, 1, 1, 0, a + 1, a, 0, a, a, 0, a, a, 1)

Passes extra positional or keyword arguments through::

sage: C.random_element(prob=.5, distribution='1/n') # random test

(1, 0, a, 0, 0, 0, 0, a + 1, 0, 0, 0, 0, 0, 0, 0, 0, a + 1, a + 1, 1, 0, 0)

TESTS:

Test that the codeword returned is immutable (see :trac:`16469`)::

sage: c = C.random_element()

sage: c.is_immutable()

True

Test that codeword returned has the same parent as any non-random codeword

(see :trac:`19653`)::

sage: C = codes.random_linear_code(GF(16, 'a'), 10, 4)

sage: c1 = C.random_element()

sage: c2 = C[1]

sage: c1.parent() == c2.parent()

True

"""

E = self.encoder()

M = E.message_space()

m = M.random_element(*args, **kwds)

c = E.encode(m)

c.set_immutable()

return c

def relative_distance(self):

r"""

Return the ratio of the minimum distance to the code length.

EXAMPLES::

sage: C = codes.HammingCode(GF(2),3)

sage: C.relative_distance()

3/7

"""

return self.minimum_distance() / self.length()

def rate(self):

r"""

Return the ratio of the number of information symbols to

the code length.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.rate()

4/7

"""

return self.dimension() / self.length()

def redundancy_matrix(self):

r"""

Returns the non-identity columns of a systematic generator matrix for

``self``.

A systematic generator matrix is a generator matrix such that a subset

of its columns forms the identity matrix. This method returns the

remaining part of the matrix.

For any given code, there can be many systematic generator matrices

(depending on which positions should form the identity). This method

will use the matrix returned by

:meth:`AbstractLinearCode.systematic_generator_matrix`.

OUTPUT:

- An `k \times (n-k)` matrix.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.generator_matrix()

[1 0 0 0 0 1 1]

[0 1 0 0 1 0 1]

[0 0 1 0 1 1 0]

[0 0 0 1 1 1 1]

sage: C.redundancy_matrix()

[0 1 1]

[1 0 1]

[1 1 0]

[1 1 1]

sage: C = LinearCode(matrix(GF(3),2,[1,2,0,\

2,1,1]))

sage: C.systematic_generator_matrix()

[1 2 0]

[0 0 1]

sage: C.redundancy_matrix()

[2]

[0]

"""

E = self.encoder("Systematic")

G = E.generator_matrix()

return G.delete_columns(E.systematic_positions())

def sd_duursma_data(self, i, warn=True):

r"""

Compute two integers pertaining to the computation of the self-dual

Duursma zeta function for ``self``, if ``self`` is a self-dual code.

INPUT:

- ``i`` - Type number

OUTPUT:

- Pair ``(v, m)`` as in Duursma [Du2003]_

EXAMPLES::

sage: MS = MatrixSpace(GF(2),2,4)

sage: G = MS([1,1,0,0,0,0,1,1])

sage: C = LinearCode(G)

sage: C == C.dual_code() # checks that C is self dual

True

sage: for i in [1,2,3,4]: print(C.sd_duursma_data(i))

doctest:...: DeprecationWarning: AbstractLinearCode.sd_duursma_data will be removed in a future release of Sage. Please use AbstractLinearCode.zeta_polynomial() to compute the Duursma zeta polynomial

See http://trac.sagemath.org/21165 for details.

(2, -1)

(2, -3)

(2, -2)

(2, -1)

"""

if warn:

deprecation(21165, "AbstractLinearCode.sd_duursma_data will be removed in a future release of Sage. Please use AbstractLinearCode.zeta_polynomial() to compute the Duursma zeta polynomial")

n = self.length()

d = self.minimum_distance()

if i == 1:

v = (n-4*d)//2 + 4

m = d-3

elif i == 2:

v = (n-6*d)//8 + 3

m = d-5

elif i == 3:

v = (n-4*d)//4 + 3

m = d-4

elif i == 4:

v = (n-3*d)//2 + 3

m = d-3

else:

raise ValueError("the type i should be 1,2,3 or 4")

return (ZZ(v),m)

def sd_duursma_q(self, i, d0, warn=True):

r"""

Compute a polynomial pertaining to the computation of the self-dual

Duursma zeta function for ``self``, if ``self`` is a self-dual code.

INPUT:

- ``i`` - Type number, one of 1,2,3,4

- ``d0`` - Divisor, the smallest integer such that each `A_i > 0` iff

`i` is divisible by `d0`

OUTPUT:

- The polynomial `Q(T)` as in Duursma [Du2003]_

EXAMPLES::

sage: C1 = codes.HammingCode(GF(2), 3)

sage: C2 = C1.extended_code(); C2

Extension of [7, 4] Hamming Code over GF(2)

sage: C2.sd_duursma_q(1,1)

doctest:...: DeprecationWarning: AbstractLinearCode.sd_duursma_q will be removed in a future release of Sage. Please use AbstractLinearCode.zeta_polynomial() to compute the Duursma zeta polynomial

See http://trac.sagemath.org/21165 for details.

2/5*T^2 + 2/5*T + 1/5

sage: C2.sd_duursma_q(3,1)

3/5*T^4 + 1/5*T^3 + 1/15*T^2 + 1/15*T + 1/15

"""

if warn:

deprecation(21165, "AbstractLinearCode.sd_duursma_q will be removed in a future release of Sage. Please use AbstractLinearCode.zeta_polynomial() to compute the Duursma zeta polynomial")

q = (self.base_ring()).order()

n = self.length()

d = self.minimum_distance()

d0 = self.divisor()

if i==1 or i==2:

if d>d0:

c0 = QQ((n-d)*rising_factorial(d-d0,d0+1)*self.weight_distribution()[d])/rising_factorial(n-d0-1,d0+2)

else:

c0 = QQ((n-d)*self.weight_distribution()[d])/rising_factorial(n-d0-1,d0+2)

if i==3 or i==4:

if d>d0:

c0 = rising_factorial(d-d0,d0+1)*self.weight_distribution()[d]/((q-1)*rising_factorial(n-d0,d0+1))

else:

c0 = self.weight_distribution()[d]/((q-1)*rising_factorial(n-d0,d0+1))

v, m = self.sd_duursma_data(i, warn=False)

if m<0 or v<0:

raise NotImplementedError("This combination of length and minimum distance is not supported.")

PR = PolynomialRing(QQ,"T")

T = PR.gen()

if i == 1:

coefs = PR(c0*(1+3*T+2*T**2)**m*(2*T**2+2*T+1)**v).list()

qc = [coefs[j]/binomial(4*m+2*v,m+j) for j in range(2*m+2*v+1)]

if i == 2:

F = ((T+1)**8+14*T**4*(T+1)**4+T**8)**v

coefs = (c0*(1+T)**m*(1+4*T+6*T**2+4*T**3)**m*F).coefficients(sparse=False)

qc = [coefs[j]/binomial(6*m+8*v,m+j) for j in range(4*m+8*v+1)]

if i == 3:

F = (3*T**2+4*T+1)**v*(1+3*T**2)**v

# Note that: (3*T**2+4*T+1)(1+3*T**2)=(T+1)**4+8*T**3*(T+1)

coefs = (c0*(1+3*T+3*T**2)**m*F).coefficients(sparse=False)

qc = [coefs[j]/binomial(4*m+4*v,m+j) for j in range(2*m+4*v+1)]

if i == 4:

coefs = (c0*(1+2*T)**m*(4*T**2+2*T+1)**v).coefficients(sparse=False)

qc = [coefs[j]/binomial(3*m+2*v,m+j) for j in range(m+2*v+1)]

Q = PR(qc)

return Q/Q(1)

def sd_zeta_polynomial(self, typ=1):

r"""

Return the Duursma zeta polynomial, computed in a fashion that only

works if ``self`` is self-dual.

.. WARNING::

This function does not check that ``self`` is self-dual. Indeed, it

is not even clear which notion of self-dual is supported ([Du2003]_ seems

to indicate formal self-dual, but the example below is a hexacode

which is Hermitian self-dual).

INPUT:

- ``typ`` - Integer, type of this s.d. code; one of 1,2,3, or

4 (default: 1)

OUTPUT:

- Polynomial in a variable "T": the Duursma zeta function as in [Du2003]_

EXAMPLES::

sage: C1 = codes.HammingCode(GF(2), 3)

sage: C2 = C1.extended_code(); C2

Extension of [7, 4] Hamming Code over GF(2)

sage: P = C2.sd_zeta_polynomial(); P

doctest:...: DeprecationWarning: AbstractLinearCode.sd_zeta_polynomial() will be removed in a future release of Sage. Please use AbstractLinearCode.zeta_polynomial() instead

See http://trac.sagemath.org/21165 for details.

2/5*T^2 + 2/5*T + 1/5

sage: P(1)

sage: F.<z> = GF(4,"z")

sage: MS = MatrixSpace(F, 3, 6)

sage: G = MS([[1,0,0,1,z,z],[0,1,0,z,1,z],[0,0,1,z,z,1]])

sage: C = LinearCode(G) # the "hexacode"

sage: C.sd_zeta_polynomial(4)

"""

deprecation(21165, "AbstractLinearCode.sd_zeta_polynomial() will be removed in a future release of Sage. Please use AbstractLinearCode.zeta_polynomial() instead")

if not self.base_field().cardinality() <= 4:

raise ValueError("the Duursma zeta polynomial is only defined for codes over GF(2), GF(3) or GF(4).")

d0 = self.divisor()

Q = self.sd_duursma_q(typ,d0, warn=False)

PR = Q.parent()

T = PR.fraction_field().gen()

if typ == 1:

P0 = Q

if typ == 2:

P0 = Q/(1-2*T+2*T**2)

if typ == 3:

P0 = Q/(1+3*T**2)

if typ == 4:

P0 = Q/(1+2*T)

return PR(P0/P0(1))

def shortened(self, L):

r"""

Returns the code shortened at the positions ``L``, where

`L \subset \{1,2,...,n\}`.

Consider the subcode `C(L)` consisting of all codewords `c\in C` which

satisfy `c_i=0` for all `i\in L`. The punctured code `C(L)^L` is

called the shortened code on `L` and is denoted `C_L`. The code

constructed is actually only isomorphic to the shortened code defined

in this way.

By Theorem 1.5.7 in [HP2003]_, `C_L` is `((C^\perp)^L)^\perp`. This is used

in the construction below.

INPUT:

- ``L`` - Subset of `\{1,...,n\}`, where `n` is the length of this code

OUTPUT:

- Linear code, the shortened code described above

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.shortened([1,2])

[5, 2] linear code over GF(2)

"""

Cd = self.dual_code()

Cdp = Cd.punctured(set(L))

return Cdp.dual_code()

@cached_method

def weight_distribution(self, algorithm=None):

r"""

Returns the weight distribution, or spectrum, of ``self`` as a list.

The weight distribution a code of length `n` is the sequence `A_0,

A_1,..., A_n` where `A_i` is the number of codewords of weight `i`.

INPUT:

- ``algorithm`` - (optional, default: ``None``) If set to ``"gap"``,

call GAP. If set to `"leon"`, call the option GAP package GUAVA and

call a function therein by Jeffrey Leon (see warning below). If set to

``"binary"``, use an algorithm optimized for binary codes. The default

is to use ``"binary"`` for binary codes and ``"gap"`` otherwise.

OUTPUT:

- A list of non-negative integers: the weight distribution.

.. WARNING::

Specifying ``algorithm = "leon"`` sometimes prints a traceback

related to a stack smashing error in the C library. The result

appears to be computed correctly, however. It appears to run much

faster than the GAP algorithm in small examples and much slower than

the GAP algorithm in larger examples.

EXAMPLES::

sage: MS = MatrixSpace(GF(2),4,7)

sage: G = MS([[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: C.weight_distribution()

[1, 0, 0, 7, 7, 0, 0, 1]

sage: F.<z> = GF(2^2,"z")

sage: C = codes.HammingCode(F, 2); C

[5, 3] Hamming Code over GF(4)

sage: C.weight_distribution()

[1, 0, 0, 30, 15, 18]

sage: C = codes.HammingCode(GF(2), 3); C

[7, 4] Hamming Code over GF(2)

sage: C.weight_distribution(algorithm="leon") # optional - gap_packages (Guava package)

[1, 0, 0, 7, 7, 0, 0, 1]

sage: C.weight_distribution(algorithm="gap")

[1, 0, 0, 7, 7, 0, 0, 1]

sage: C.weight_distribution(algorithm="binary")

[1, 0, 0, 7, 7, 0, 0, 1]

sage: C = codes.HammingCode(GF(3), 3); C

[13, 10] Hamming Code over GF(3)

sage: C.weight_distribution() == C.weight_distribution(algorithm="leon") # optional - gap_packages (Guava package)

True

sage: C = codes.HammingCode(GF(5), 2); C

[6, 4] Hamming Code over GF(5)

sage: C.weight_distribution() == C.weight_distribution(algorithm="leon") # optional - gap_packages (Guava package)

True

sage: C = codes.HammingCode(GF(7), 2); C

[8, 6] Hamming Code over GF(7)

sage: C.weight_distribution() == C.weight_distribution(algorithm="leon") # optional - gap_packages (Guava package)

True

"""

if algorithm is None:

if self.base_ring().order() == 2:

algorithm = "binary"

else:

algorithm = "gap"

F = self.base_ring()

n = self.length()

G = self.generator_matrix()

if algorithm=="gap":

Gmat = self.generator_matrix()._gap_init_()

G = gap(Gmat)

q = self.base_ring().order()

k = gap(self.base_ring())

z = 'Z(%s)*%s'%(q, [0]*self.length()) # GAP zero vector as a string

_ = gap.eval("w:=DistancesDistributionMatFFEVecFFE("+Gmat+", GF("+str(q)+"),"+z+")")

v = [eval(gap.eval("w["+str(i)+"]")) for i in range(1,self.length()+2)] # because GAP returns vectors in compressed form

return v

elif algorithm=="binary":

from sage.coding.binary_code import weight_dist

return weight_dist(self.generator_matrix())

elif algorithm=="leon":

if not(F.order() in [2,3,5,7]):

raise NotImplementedError("The algorithm 'leon' is only implemented for q = 2,3,5,7.")

# The GAP command DirectoriesPackageLibrary tells the location of the latest

# version of the Guava libraries, so gives us the location of the Guava binaries too.

guava_bin_dir = gap.eval('DirectoriesPackagePrograms("guava")[1]')

guava_bin_dir = guava_bin_dir[guava_bin_dir.index('"') + 1:guava_bin_dir.rindex('"')]

input = _dump_code_in_leon_format(self) + "::code"

import os

import subprocess

lines = subprocess.check_output([os.path.join(guava_bin_dir, 'wtdist'), input])

from six import StringIO

# to use the already present output parser

wts = [0] * (n + 1)

s = 0

for L in StringIO(lines).readlines():

L = L.strip()

if L:

o = ord(L[0])

if o >= 48 and o <= 57:

wt, num = L.split()

wts[eval(wt)] = eval(num)

return wts

else:

raise NotImplementedError("The only algorithms implemented currently are 'gap', 'leon' and 'binary'.")

spectrum = weight_distribution

def standard_form(self, return_permutation=True):

r"""

Returns a linear code which is permutation-equivalent to ``self`` and

admits a generator matrix in standard form.

A generator matrix is in standard form if it is of the form `[I \vert

A]`, where `I` is the `k \times k` identity matrix. Any code admits a

generator matrix in systematic form, i.e. where a subset of the columns

form the identity matrix, but one might need to permute columns to allow

the identity matrix to be leading.

INPUT:

- ``return_permutation`` -- (default: ``True``) if ``True``, the column

permutation which brings ``self`` into the returned code is also

returned.

OUTPUT:

- A :class:`LinearCode` whose :meth:`systematic_generator_matrix` is

guaranteed to be of the form `[I \vert A]`.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.generator_matrix()

[1 0 0 0 0 1 1]

[0 1 0 0 1 0 1]

[0 0 1 0 1 1 0]

[0 0 0 1 1 1 1]

sage: Cs,p = C.standard_form()

sage: p

[]

sage: Cs is C

True

sage: C = LinearCode(matrix(GF(2), [[1,0,0,0,1,1,0],\

[0,1,0,1,0,1,0],\

[0,0,0,0,0,0,1]]))

sage: Cs, p = C.standard_form()

sage: p

[1, 2, 7, 3, 4, 5, 6]

sage: Cs.generator_matrix()

[1 0 0 0 0 1 1]

[0 1 0 0 1 0 1]

[0 0 1 0 0 0 0]

"""

E = self.encoder("Systematic")

if E.systematic_positions() == tuple(range(self.dimension())):

from sage.combinat.permutation import Permutation

return self, Permutation([])

else:

perm = E.systematic_permutation()

return self.permuted_code(perm), perm

def support(self):

r"""

Returns the set of indices `j` where `A_j` is nonzero, where

`A_j` is the number of codewords in `self` of Hamming weight `j`.

OUTPUT:

- List of integers

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.weight_distribution()

[1, 0, 0, 7, 7, 0, 0, 1]

sage: C.support()

[0, 3, 4, 7]

"""

n = self.length()

F = self.base_ring()

V = VectorSpace(F,n+1)

return V(self.weight_distribution()).support()

def syndrome(self, r):

r"""

Returns the syndrome of ``r``.

The syndrome of ``r`` is the result of `H \times r` where `H` is

the parity check matrix of ``self``. If ``r`` belongs to ``self``,

its syndrome equals to the zero vector.

INPUT:

- ``r`` -- a vector of the same length as ``self``

OUTPUT:

- a column vector

EXAMPLES::

sage: MS = MatrixSpace(GF(2),4,7)

sage: G = MS([[1,1,1,0,0,0,0], [1,0,0,1,1,0,0], [0,1,0,1,0,1,0], [1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: r = vector(GF(2), (1,0,1,0,1,0,1))

sage: r in C

True

sage: C.syndrome(r)

(0, 0, 0)

If ``r`` is not a codeword, its syndrome is not equal to zero::

sage: r = vector(GF(2), (1,0,1,0,1,1,1))

sage: r in C

False

sage: C.syndrome(r)

(0, 1, 1)

Syndrome computation works fine on bigger fields::

sage: C = codes.random_linear_code(GF(59), 12, 4)

sage: c = C.random_element()

sage: C.syndrome(c)

(0, 0, 0, 0, 0, 0, 0, 0)

"""

return self.parity_check_matrix()*r

def unencode(self, c, encoder_name=None, nocheck=False, **kwargs):

r"""

Returns the message corresponding to ``c``.

This is the inverse of :meth:`encode`.

INPUT:

- ``c`` -- a codeword of ``self``.

- ``encoder_name`` -- (default: ``None``) name of the decoder which will be used

to decode ``word``. The default decoder of ``self`` will be used if

default value is kept.

- ``nocheck`` -- (default: ``False``) checks if ``c`` is in ``self``. You might set

this to ``True`` to disable the check for saving computation. Note that if ``c`` is

not in ``self`` and ``nocheck = True``, then the output of :meth:`unencode` is

not defined (except that it will be in the message space of ``self``).

- ``kwargs`` -- all additional arguments are forwarded to the construction of the

encoder that is used.

OUTPUT:

- an element of the message space of ``encoder_name`` of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: c = vector(GF(2), (1, 1, 0, 0, 1, 1, 0))

sage: C.unencode(c)

(0, 1, 1, 0)

"""

E = self.encoder(encoder_name, **kwargs)

return E.unencode(c, nocheck)

def weight_enumerator(self, names=None, name2=None, bivariate=True):

"""

Return the weight enumerator polynomial of ``self``.

This is the bivariate, homogeneous polynomial in `x` and `y` whose

coefficient to `x^i y^{n-i}` is the number of codewords of `self` of

Hamming weight `i`. Here, `n` is the length of `self`.

INPUT:

- ``names`` - (default: ``"xy"``) The names of the variables in the

homogeneous polynomial. Can be given as a single string of length 2,

or a single string with a comma, or as a tuple or list of two strings.

- ``name2`` - Deprecated, (default: ``None``) The string name of the

second variable.

- ``bivariate`` - (default: `True`) Whether to return a bivariate,

homogeneous polynomial or just a univariate polynomial. If set to

``False``, then ``names`` will be interpreted as a single variable

name and default to ``"x"``.

OUTPUT:

- The weight enumerator polynomial over `\ZZ`.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.weight_enumerator()

x^7 + 7*x^4*y^3 + 7*x^3*y^4 + y^7

sage: C.weight_enumerator(names="st")

s^7 + 7*s^4*t^3 + 7*s^3*t^4 + t^7

sage: C.weight_enumerator(names="var1, var2")

var1^7 + 7*var1^4*var2^3 + 7*var1^3*var2^4 + var2^7

sage: C.weight_enumerator(names=('var1', 'var2'))

var1^7 + 7*var1^4*var2^3 + 7*var1^3*var2^4 + var2^7

sage: C.weight_enumerator(bivariate=False)

x^7 + 7*x^4 + 7*x^3 + 1

An example of a code with a non-symmetrical weight enumerator::

sage: C = codes.GolayCode(GF(3), extended=False)

sage: C.weight_enumerator()

24*x^11 + 110*x^9*y^2 + 330*x^8*y^3 + 132*x^6*y^5 + 132*x^5*y^6 + y^11

"""

if names is None:

if bivariate:

names = "xy"

else:

names = "x"

else:

if name2 is not None:

from sage.misc.superseded import deprecation

deprecation(21576, "Optional argument name2 is deprecated. You should just give a tuple to `names`.")

names = (names, name2)

spec = self.weight_distribution()

n = self.length()

if bivariate:

R = PolynomialRing(ZZ,2,names)

x,y = R.gens()

return sum(spec[i]*x**i*y**(n-i) for i in range(n+1))

else:

R = PolynomialRing(ZZ,names)

x, = R.gens()

return sum(spec[i]*x**i for i in range(n+1))

@cached_method

def zero(self):

r"""

Returns the zero vector of ``self``.

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.zero()

(0, 0, 0, 0, 0, 0, 0)

sage: C.sum(()) # indirect doctest

(0, 0, 0, 0, 0, 0, 0)

sage: C.sum((C.gens())) # indirect doctest

(1, 1, 1, 1, 1, 1, 1)

"""

return self.ambient_space().zero()

def zeta_polynomial(self, name="T"):

r"""

Returns the Duursma zeta polynomial of this code.

Assumes that the minimum distances of this code and its dual are

greater than 1. Prints a warning to ``stdout`` otherwise.

INPUT:

- ``name`` - String, variable name (default: ``"T"``)

OUTPUT:

- Polynomial over `\QQ`

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.zeta_polynomial()

2/5*T^2 + 2/5*T + 1/5

sage: C = codes.databases.best_linear_code_in_guava(6,3,GF(2)) # optional - gap_packages (Guava package)

sage: C.minimum_distance() # optional - gap_packages (Guava package)

sage: C.zeta_polynomial() # optional - gap_packages (Guava package)

2/5*T^2 + 2/5*T + 1/5

sage: C = codes.HammingCode(GF(2), 4)

sage: C.zeta_polynomial()

16/429*T^6 + 16/143*T^5 + 80/429*T^4 + 32/143*T^3 + 30/143*T^2 + 2/13*T + 1/13

sage: F.<z> = GF(4,"z")

sage: MS = MatrixSpace(F, 3, 6)

sage: G = MS([[1,0,0,1,z,z],[0,1,0,z,1,z],[0,0,1,z,z,1]])

sage: C = LinearCode(G) # the "hexacode"

sage: C.zeta_polynomial()

REFERENCES:

.. [Du01] \I. Duursma, "From weight enumerators to zeta functions", in

Discrete Applied Mathematics, vol. 111, no. 1-2, pp. 55-73, 2001.

"""

n = self.length()

q = (self.base_ring()).order()

d = self.minimum_distance()

dperp = (self.dual_code()).minimum_distance()

if d == 1 or dperp == 1:

print("\n WARNING: There is no guarantee this function works when the minimum distance")

print(" of the code or of the dual code equals 1.\n")

RT = PolynomialRing(QQ,"%s"%name)

R = PolynomialRing(QQ,3,"xy%s"%name)

x,y,T = R.gens()

we = self.weight_enumerator()

A = R(we)

#B = A(x+y,y,T)-(x+y)**n

B = A(x,x+y,T)-(x+y)**n

Bs = B.coefficients()

Bs.reverse()

b = [Bs[i]/binomial(n,i+d) for i in range(len(Bs))]

r = n-d-dperp+2

P_coeffs = []

for i in range(len(b)):

if i == 0:

P_coeffs.append(b[0])

if i == 1:

P_coeffs.append(b[1] - (q+1)*b[0])

if i>1:

P_coeffs.append(b[i] - (q+1)*b[i-1] + q*b[i-2])

P = sum([P_coeffs[i]*T**i for i in range(r+1)])

return RT(P)/RT(P)(1)

def zeta_function(self, name="T"):

r"""

Returns the Duursma zeta function of the code.

INPUT:

- ``name`` - String, variable name (default: ``"T"``)

OUTPUT:

- Element of `\QQ(T)`

EXAMPLES::

sage: C = codes.HammingCode(GF(2), 3)

sage: C.zeta_function()

(2/5*T^2 + 2/5*T + 1/5)/(2*T^2 - 3*T + 1)

"""

P = self.zeta_polynomial()

q = (self.base_ring()).characteristic()

RT = PolynomialRing(QQ,"%s"%name)

T = RT.gen()

return P/((1-T)*(1-q*T))

def LinearCodeFromVectorSpace(V, d=None):

"""

Simply converts a vector subspace `V` of `GF(q)^n` into a `LinearCode`.

INPUT:

- ``V`` -- The vector space

- ``d`` -- (Optional, default: ``None``) the minimum distance of the

code, if known. This is an optional parameter.

.. note::

The veracity of the minimum distance ``d``, if provided, is not

checked.

EXAMPLES::

sage: V = VectorSpace(GF(2), 8)

sage: L = V.subspace([[1,1,1,1,0,0,0,0],[0,0,0,0,1,1,1,1]])

sage: C = LinearCodeFromVectorSpace(L)

doctest:...: DeprecationWarning: LinearCodeFromVectorSpace is deprecated. Simply call LinearCode with your vector space instead.

See http://trac.sagemath.org/21165 for details.

sage: C.generator_matrix()

[1 1 1 1 0 0 0 0]

[0 0 0 0 1 1 1 1]

sage: C.minimum_distance()

Here, we provide the minimum distance of the code.::

sage: C = LinearCodeFromVectorSpace(L, d=4)

sage: C.minimum_distance()

"""

from sage.misc.superseded import deprecation

deprecation(21165, "LinearCodeFromVectorSpace is deprecated. Simply call LinearCode with your vector space instead.")

F = V.base_ring()

B = V.basis()

n = len(B[0].list())

k = len(B)

MS = MatrixSpace(F,k,n)

G = MS([B[i].list() for i in range(k)])

return LinearCode(G, d=d)

############################ linear codes python class ########################

class LinearCode(AbstractLinearCode):

r"""

Linear codes over a finite field or finite ring, represented using a

generator matrix.

This class should be used for arbitrary and unstructured linear codes. This

means that basic operations on the code, such as the computation of the

minimum distance, will use generic, slow algorithms.

If you are looking for constructing a code from a more specific family, see

if the family has been implemented by investigating `codes.<tab>`. These

more specific classes use properties particular to that family to allow

faster algorithms, and could also have family-specific methods.

See :wikipedia:`Linear_code` for more information on unstructured linear codes.

INPUT:

- ``generator`` -- a generator matrix over a finite field (``G`` can be

defined over a finite ring but the matrices over that ring must have

certain attributes, such as ``rank``); or a code over a finite field

- ``d`` -- (optional, default: ``None``) the minimum distance of the code

.. NOTE::

The veracity of the minimum distance ``d``, if provided, is not

checked.

EXAMPLES::

sage: MS = MatrixSpace(GF(2),4,7)

sage: G = MS([[1,1,1,0,0,0,0], [1,0,0,1,1,0,0], [0,1,0,1,0,1,0], [1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: C

[7, 4] linear code over GF(2)

sage: C.base_ring()

Finite Field of size 2

sage: C.dimension()

sage: C.length()

sage: C.minimum_distance()

sage: C.spectrum()

[1, 0, 0, 7, 7, 0, 0, 1]

sage: C.weight_distribution()

[1, 0, 0, 7, 7, 0, 0, 1]

The minimum distance of the code, if known, can be provided as an

optional parameter.::

sage: C = LinearCode(G, d=3)

sage: C.minimum_distance()

Another example.::

sage: MS = MatrixSpace(GF(5),4,7)

sage: G = MS([[1,1,1,0,0,0,0], [1,0,0,1,1,0,0], [0,1,0,1,0,1,0], [1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: C

[7, 4] linear code over GF(5)

Providing a code as the parameter in order to "forget" its structure (see

:trac:`20198`)::

sage: C = codes.GeneralizedReedSolomonCode(GF(23).list(), 12)

sage: LinearCode(C)

[23, 12] linear code over GF(23)

Another example::

sage: C = codes.HammingCode(GF(7), 3)

sage: C

[57, 54] Hamming Code over GF(7)

sage: LinearCode(C)

[57, 54] linear code over GF(7)

AUTHORS:

- David Joyner (11-2005)

- Charles Prior (03-2016): :trac:`20198`, LinearCode from a code

"""

def __init__(self, generator, d=None):

r"""

See the docstring for :meth:`LinearCode`.

EXAMPLES::

sage: MS = MatrixSpace(GF(2),4,7)

sage: G = MS([[1,1,1,0,0,0,0], [1,0,0,1,1,0,0], [0,1,0,1,0,1,0], [1,1,0,1,0,0,1]])

sage: C = LinearCode(G) # indirect doctest

sage: C

[7, 4] linear code over GF(2)

The minimum distance of the code, if known, can be provided as an

optional parameter.::

sage: C = LinearCode(G, d=3)

sage: C.minimum_distance()

TESTS::

sage: C = codes.HammingCode(GF(2), 3)

sage: TestSuite(C).run()

Check that it works even with input matrix with non full rank (see

:trac:`17452`)::

sage: K.<a> = GF(4)

sage: G = matrix([[a, a + 1, 1, a + 1, 1, 0, 0],

....: [0, a, a + 1, 1, a + 1, 1, 0],

....: [0, 0, a, a + 1, 1, a + 1, 1],

....: [a + 1, 0, 1, 0, a + 1, 1, a + 1],

....: [a, a + 1, a + 1, 0, 0, a + 1, 1],

....: [a + 1, a, a, 1, 0, 0, a + 1],

....: [a, a + 1, 1, a + 1, 1, 0, 0]])

sage: C = LinearCode(G)

sage: C.basis()

[

(1, 0, 0, a + 1, 0, 1, 0),

(0, 1, 0, 0, a + 1, 0, 1),

(0, 0, 1, a, a + 1, a, a + 1)

]

sage: C.minimum_distance()

We can construct a linear code directly from a vector space

sage: VS = matrix(GF(2), [[1,0,1],\

[1,0,1]]).row_space()

sage: C = LinearCode(VS); C

[3, 1] linear code over GF(2)

Forbid the zero vector space (see :trac:`17452` and :trac:`6486`)::

sage: G = matrix(GF(2), [[0,0,0]])

sage: C = LinearCode(G)

Traceback (most recent call last):

...

ValueError: this linear code contains no non-zero vector

"""

base_ring = generator.base_ring()

if not base_ring.is_field():

raise ValueError("'generator' must be defined on a field (not a ring)")

try:

basis = None

if hasattr(generator,"nrows"): # generator matrix case

if generator.rank() < generator.nrows():

basis = generator.row_space().basis()

else:

basis = generator.basis() # vector space etc. case

if not basis is None:

from sage.matrix.constructor import matrix

generator = matrix(base_ring, basis)

if generator.nrows() == 0:

raise ValueError("this linear code contains no non-zero vector")

except AttributeError:

# Assume input is an AbstractLinearCode, extract its generator matrix

generator = generator.generator_matrix()

super(LinearCode, self).__init__(base_ring, generator.ncols(), "GeneratorMatrix", "Syndrome")

self._generator_matrix = generator

self._dimension = generator.rank()

self._minimum_distance = d

def _repr_(self):

r"""

See the docstring for :meth:`LinearCode`.

EXAMPLES::

sage: MS = MatrixSpace(GF(2),4,7)

sage: G = MS([[1,1,1,0,0,0,0], [1,0,0,1,1,0,0], [0,1,0,1,0,1,0], [1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: C # indirect doctest

[7, 4] linear code over GF(2)

"""

R = self.base_ring()

if R in Fields():

return "[%s, %s] linear code over GF(%s)"%(self.length(), self.dimension(), R.cardinality())

else:

return "[%s, %s] linear code over %s"%(self.length(), self.dimension(), R)

def _latex_(self):

"""

Return a latex representation of ``self``.

EXAMPLES::

sage: MS = MatrixSpace(GF(2),4,7)

sage: G = MS([[1,1,1,0,0,0,0], [1,0,0,1,1,0,0], [0,1,0,1,0,1,0], [1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: latex(C)

[7, 4]\textnormal{ Linear code over }\Bold{F}_{2}

"""

return "[%s, %s]\\textnormal{ Linear code over }%s"\

% (self.length(), self.dimension(), self.base_ring()._latex_())

def __hash__(self):

r"""

Returns the hash value of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: hash(C) #random

9015017528451745710

If ``C1`` and ``C2`` are two codes which only differ by the

coefficients of their generator matrices, their hashes are

different (we check that the bug found in :trac:`18813` is

fixed)::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C1 = LinearCode(G)

sage: G = Matrix(GF(2), [[1,0,0,1,0,1,0],[0,1,0,0,1,0,0],[0,0,1,1,0,1,0],[0,0,0,0,0,0,1]])

sage: C2 = LinearCode(G)

sage: hash(C1) != hash(C2)

True

"""

Str = str(self)

G = self.generator_matrix()

return hash((Str, G)) ^ hash(Str) ^ hash(G)

def generator_matrix(self, encoder_name=None, **kwargs):

r"""

Returns a generator matrix of ``self``.

INPUT:

- ``encoder_name`` -- (default: ``None``) name of the encoder which will be

used to compute the generator matrix. ``self._generator_matrix``

will be returned if default value is kept.

- ``kwargs`` -- all additional arguments are forwarded to the construction of the

encoder that is used.

EXAMPLES::

sage: G = matrix(GF(3),2,[1,-1,1,-1,1,1])

sage: code = LinearCode(G)

sage: code.generator_matrix()

[1 2 1]

[2 1 1]

"""

if encoder_name is None or encoder_name is 'GeneratorMatrix':

g = self._generator_matrix

else:

g = super(LinearCode, self).generator_matrix(encoder_name, **kwargs)

g.set_immutable()

return g

####################### encoders ###############################

class LinearCodeGeneratorMatrixEncoder(Encoder):

r"""

Encoder based on generator_matrix for Linear codes.

This is the default encoder of a generic linear code, and should never be used for other codes than

:class:`LinearCode`.

INPUT:

- ``code`` -- The associated :class:`LinearCode` of this encoder.

"""

def __init__(self, code):

r"""

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeGeneratorMatrixEncoder(C)

sage: E

Generator matrix-based encoder for [7, 4] linear code over GF(2)

"""

super(LinearCodeGeneratorMatrixEncoder, self).__init__(code)

def __eq__(self, other):

r"""

Tests equality between LinearCodeGeneratorMatrixEncoder objects.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: E1 = LinearCode(G).encoder()

sage: E2 = LinearCode(G).encoder()

sage: E1 == E2

True

"""

return isinstance(other, LinearCodeGeneratorMatrixEncoder)\

and self.code() == other.code()

def _repr_(self):

r"""

Returns a string representation of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeGeneratorMatrixEncoder(C)

sage: E

Generator matrix-based encoder for [7, 4] linear code over GF(2)

"""

return "Generator matrix-based encoder for %s" % self.code()

def _latex_(self):

r"""

Returns a latex representation of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeGeneratorMatrixEncoder(C)

sage: latex(E)

\textnormal{Generator matrix-based encoder for }[7, 4]\textnormal{ Linear code over }\Bold{F}_{2}

"""

return "\\textnormal{Generator matrix-based encoder for }%s" % self.code()._latex_()

@cached_method

def generator_matrix(self):

r"""

Returns a generator matrix of the associated code of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeGeneratorMatrixEncoder(C)

sage: E.generator_matrix()

[1 1 1 0 0 0 0]

[1 0 0 1 1 0 0]

[0 1 0 1 0 1 0]

[1 1 0 1 0 0 1]

"""

g = self.code().generator_matrix()

g.set_immutable()

return g

class LinearCodeParityCheckEncoder(Encoder):

r"""

Encoder based on :meth:`parity_check_matrix` for Linear codes.

It constructs the generator matrix through the parity check matrix.

INPUT:

- ``code`` -- The associated code of this encoder.

"""

def __init__(self, code):

r"""

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeParityCheckEncoder(C)

doctest:...: DeprecationWarning: LinearCodeParityCheckEncoder is now deprecated. Please use LinearCodeSystematicEncoder instead.

See http://trac.sagemath.org/20835 for details.

sage: E

Parity check matrix-based encoder for [7, 4] linear code over GF(2)

"""

from sage.misc.superseded import deprecation

deprecation(20835, "LinearCodeParityCheckEncoder is now deprecated. Please use LinearCodeSystematicEncoder instead.")

super(LinearCodeParityCheckEncoder, self).__init__(code)

def _repr_(self):

r"""

Return a string representation of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeParityCheckEncoder(C)

doctest:...: DeprecationWarning: LinearCodeParityCheckEncoder is now deprecated. Please use LinearCodeSystematicEncoder instead.

See http://trac.sagemath.org/20835 for details.

sage: E

Parity check matrix-based encoder for [7, 4] linear code over GF(2)

"""

return "Parity check matrix-based encoder for %s" % self.code()

def _latex_(self):

r"""

Return a latex representation of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeParityCheckEncoder(C)

doctest:...: DeprecationWarning: LinearCodeParityCheckEncoder is now deprecated. Please use LinearCodeSystematicEncoder instead.

See http://trac.sagemath.org/20835 for details.

sage: latex(E)

\textnormal{Parity check matrix-based encoder for }[7, 4]\textnormal{ Linear code over }\Bold{F}_{2}

"""

return "\\textnormal{Parity check matrix-based encoder for }%s" % self.code()._latex_()

@cached_method

def generator_matrix(self):

r"""

Returns a generator matrix of the associated code of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeParityCheckEncoder(C)

doctest:...: DeprecationWarning: LinearCodeParityCheckEncoder is now deprecated. Please use LinearCodeSystematicEncoder instead.

See http://trac.sagemath.org/20835 for details.

sage: E.generator_matrix()

[1 0 0 0 0 1 1]

[0 1 0 0 1 0 1]

[0 0 1 0 1 1 0]

[0 0 0 1 1 1 1]

"""

g = self.code().parity_check_matrix().right_kernel_matrix()

g.set_immutable()

return g

class LinearCodeSystematicEncoder(Encoder):

r"""

Encoder based on a generator matrix in systematic form for Linear codes.

To encode an element of its message space, this encoder first builds a

generator matrix in systematic form. What is called systematic form here

is the reduced row echelon form of a matrix, which is not necessarily

`[I \vert H]`, where `I` is the identity block and `H` the parity block.

One can refer to :meth:`LinearCodeSystematicEncoder.generator_matrix`

for a concrete example.

Once such a matrix has been computed, it is used to encode any message

into a codeword.

This encoder can also serve as the default encoder of a code defined by a

parity check matrix: if the :class:`LinearCodeSystematicEncoder` detects

that it is the default encoder, it computes a generator matrix as the

reduced row echelon form of the right kernel of the parity check matrix.

INPUT:

- ``code`` -- The associated code of this encoder.

- ``systematic_positions`` -- (default: ``None``) the positions in codewords that

should correspond to the message symbols. A list of `k` distinct integers in

the range 0 to `n-1` where `n` is the length of the code and `k` its

dimension. The 0th symbol of a message will then be at position

``systematic_positions[0]``, the 1st index at position

``systematic_positions[1]``, etc. A ``ValueError`` is raised at

construction time if the supplied indices do not form an information set.

EXAMPLES:

The following demonstrates the basic usage of :class:`LinearCodeSystematicEncoder`::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0,0],\

[1,0,0,1,1,0,0,0],\

[0,1,0,1,0,1,0,0],\

[1,1,0,1,0,0,1,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeSystematicEncoder(C)

sage: E.generator_matrix()

[1 0 0 0 0 1 1 1]

[0 1 0 0 1 0 1 1]

[0 0 1 0 1 1 0 0]

[0 0 0 1 1 1 1 1]

sage: E2 = codes.encoders.LinearCodeSystematicEncoder(C, systematic_positions=[5,4,3,2])

sage: E2.generator_matrix()

[1 0 0 0 0 1 1 1]

[0 1 0 0 1 0 1 1]

[1 1 0 1 0 0 1 1]

[1 1 1 0 0 0 0 0]

An error is raised if one specifies systematic positions which do not form

an information set::

sage: E3 = codes.encoders.LinearCodeSystematicEncoder(C, systematic_positions=[0,1,6,7])

Traceback (most recent call last):

...

ValueError: systematic_positions are not an information set

We exemplify how to use :class:`LinearCodeSystematicEncoder` as the default

encoder. The following class is the dual of the repetition code::

sage: class DualRepetitionCode(sage.coding.linear_code.AbstractLinearCode):

....: def __init__(self, field, length):

....: sage.coding.linear_code.AbstractLinearCode.__init__(self,field, length, "Systematic", "Syndrome")

....:

....: def parity_check_matrix(self):

....: return Matrix(self.base_field(), [1]*self.length())

....:

....: def _repr_(self):

....: return "Dual of the [%d, 1] Repetition Code over GF(%s)" % (self.length(), self.base_field().cardinality())

....:

sage: DualRepetitionCode(GF(3), 5).generator_matrix()

[1 0 0 0 2]

[0 1 0 0 2]

[0 0 1 0 2]

[0 0 0 1 2]

An exception is thrown if :class:`LinearCodeSystematicEncoder` is the default encoder but no

parity check matrix has been specified for the code::

sage: class BadCodeFamily(sage.coding.linear_code.AbstractLinearCode):

....: def __init__(self, field, length):

....: sage.coding.linear_code.AbstractLinearCode.__init__(self, field, length, "Systematic", "Syndrome")

....:

....: def _repr_(self):

....: return "I am a badly defined code"

....:

sage: BadCodeFamily(GF(3), 5).generator_matrix()

Traceback (most recent call last):

...

ValueError: a parity check matrix must be specified if LinearCodeSystematicEncoder is the default encoder

"""

def __init__(self, code, systematic_positions=None):

r"""

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeSystematicEncoder(C)

sage: E

Systematic encoder for [7, 4] linear code over GF(2)

"""

super(LinearCodeSystematicEncoder, self).__init__(code)

self._systematic_positions = tuple(systematic_positions) if systematic_positions else None

if systematic_positions:

# Test that systematic_positions consists of integers in the right

# range. We test that len(systematic_positions) = code.dimension()

# in self.generator_matrix() to avoid possible infinite recursion.

if (not all( e in ZZ and e >= 0 and e < code.length() for e in systematic_positions)) \

or len(systematic_positions) != len(set(systematic_positions)):

raise ValueError("systematic positions must be a tuple of distinct integers in the range 0 to n-1 where n is the length of the code")

# Test that the systematic positions are an information set

self.generator_matrix()

def __eq__(self, other):

r"""

Tests equality between LinearCodeSystematicEncoder objects.

EXAMPLES::

sage: G = Matrix(GF(3), [[1,0,0,1,0,1,0,1,2],[0,1,0,2,2,0,1,1,0],[0,0,1,0,2,2,2,1,2]])

sage: E1 = codes.encoders.LinearCodeSystematicEncoder(LinearCode(G))

sage: E2 = codes.encoders.LinearCodeSystematicEncoder(LinearCode(G))

sage: E1 == E2

True

sage: E1.systematic_positions()

(0, 1, 2)

sage: E3 = codes.encoders.LinearCodeSystematicEncoder(LinearCode(G), systematic_positions=(2,5,6))

sage: E3.systematic_positions()

(2, 5, 6)

sage: E1 == E3

False

"""

return isinstance(other, LinearCodeSystematicEncoder)\

and self.code() == other.code()\

and self.systematic_positions() == other.systematic_positions()

def _repr_(self):

r"""

Return a string representation of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeSystematicEncoder(C)

sage: E

Systematic encoder for [7, 4] linear code over GF(2)

"""

return "Systematic encoder for %s" % self.code()

def _latex_(self):

r"""

Return a latex representation of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeSystematicEncoder(C)

sage: latex(E)

\textnormal{Systematic encoder for }[7, 4]\textnormal{ Linear code over }\Bold{F}_{2}

"""

return "\\textnormal{Systematic encoder for }%s" % self.code()._latex_()

@cached_method

def generator_matrix(self):

r"""

Returns a generator matrix in systematic form of the associated code of ``self``.

Systematic form here means that a subsets of the columns of the matrix

forms the identity matrix.

.. NOTE::

The matrix returned by this method will not necessarily be `[I \vert H]`, where `I`

is the identity block and `H` the parity block. If one wants to know which columns

create the identity block, one can call :meth:`systematic_positions`

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],\

[1,0,0,1,1,0,0],\

[0,1,0,1,0,1,0],\

[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeSystematicEncoder(C)

sage: E.generator_matrix()

[1 0 0 0 0 1 1]

[0 1 0 0 1 0 1]

[0 0 1 0 1 1 0]

[0 0 0 1 1 1 1]

We can ask for different systematic positions::

sage: E2 = codes.encoders.LinearCodeSystematicEncoder(C, systematic_positions=[5,4,3,2])

sage: E2.generator_matrix()

[1 0 0 0 0 1 1]

[0 1 0 0 1 0 1]

[1 1 0 1 0 0 1]

[1 1 1 0 0 0 0]

Another example where there is no generator matrix of the form `[I \vert H]`::

sage: G = Matrix(GF(2), [[1,1,0,0,1,0,1],\

[1,1,0,0,1,0,0],\

[0,0,1,0,0,1,0],\

[0,0,1,0,1,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeSystematicEncoder(C)

sage: E.generator_matrix()

[1 1 0 0 0 1 0]

[0 0 1 0 0 1 0]

[0 0 0 0 1 1 0]

[0 0 0 0 0 0 1]

"""

C = self.code()

# This if statement detects if this encoder is itself the default encoder.

# In this case, attempt building the generator matrix from the parity

# check matrix

if hasattr(self, "_use_pc_matrix"):

if self._use_pc_matrix == 1:

self._use_pc_matrix = 2

return C.parity_check_matrix().right_kernel_matrix()

else:

raise ValueError("a parity check matrix must be specified if LinearCodeSystematicEncoder is the default encoder")

else:

self._use_pc_matrix = 1

M = copy(C.generator_matrix())

if not self._systematic_positions:

M.echelonize()

else:

k = M.nrows() # it is important that k is *not* computed as C.dimension() to avoid possible cyclic dependency

if len(self._systematic_positions) != k:

raise ValueError("systematic_positions must be a tuple of length equal to the dimension of the code")

# Permute the columns of M and bring to reduced row echelon formb

perm = self.systematic_permutation()

M.permute_columns(perm)

M.echelonize()

if M[:,:k].is_singular():

raise ValueError("systematic_positions are not an information set")

M.permute_columns(perm.inverse())

M.set_immutable()

return M

def systematic_permutation(self):

r"""

Returns a permutation which would take the systematic positions into [0,..,k-1]

EXAMPLES::

sage: C = LinearCode(matrix(GF(2), [[1,0,0,0,1,1,0],\

[0,1,0,1,0,1,0],\

[0,0,0,0,0,0,1]]))

sage: E = codes.encoders.LinearCodeSystematicEncoder(C)

sage: E.systematic_positions()

(0, 1, 6)

sage: E.systematic_permutation()

[1, 2, 7, 3, 4, 5, 6]

"""

n = self.code().length()

systematic_positions = self.systematic_positions()

k = len(systematic_positions)

lp = [ None ]*n

for (i,j) in zip(range(k), systematic_positions):

lp[i] = j

j = k

set_sys_pos = set(systematic_positions)

for i in range(n):

if not i in set_sys_pos:

lp[j] = i

j += 1

from sage.combinat.permutation import Permutation

return Permutation([1 + e for e in lp])

def systematic_positions(self):

r"""

Returns a tuple containing the indices of the columns which form an

identity matrix when the generator matrix is in systematic form.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],\

[1,0,0,1,1,0,0],\

[0,1,0,1,0,1,0],\

[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeSystematicEncoder(C)

sage: E.systematic_positions()

(0, 1, 2, 3)

We take another matrix with a less nice shape::

sage: G = Matrix(GF(2), [[1,1,0,0,1,0,1],\

[1,1,0,0,1,0,0],\

[0,0,1,0,0,1,0],\

[0,0,1,0,1,0,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeSystematicEncoder(C)

sage: E.systematic_positions()

(0, 2, 4, 6)

The systematic positions correspond to the positions which carry information in a codeword::

sage: MS = E.message_space()

sage: m = MS.random_element()

sage: c = m * E.generator_matrix()

sage: pos = E.systematic_positions()

sage: info = MS([c[i] for i in pos])

sage: m == info

True

When constructing a systematic encoder with specific systematic

positions, then it is guaranteed that this method returns exactly those

positions (even if another choice might also be systematic)::

sage: G = Matrix(GF(2), [[1,0,0,0],\

[0,1,0,0],\

[0,0,1,1]])

sage: C = LinearCode(G)

sage: E = codes.encoders.LinearCodeSystematicEncoder(C, systematic_positions=[0,1,3])

sage: E.systematic_positions()

(0, 1, 3)

"""

return self._systematic_positions if self._systematic_positions else self.generator_matrix().pivots()

####################### decoders ###############################

class LinearCodeSyndromeDecoder(Decoder):

r"""

Constructs a decoder for Linear Codes based on syndrome lookup table.

The decoding algorithm works as follows:

- First, a lookup table is built by computing the syndrome of every error

pattern of weight up to ``maximum_error_weight``.

- Then, whenever one tries to decode a word ``r``, the syndrome of ``r`` is

computed. The corresponding error pattern is recovered from the

pre-computed lookup table.

- Finally, the recovered error pattern is subtracted from ``r`` to recover

the original word.

``maximum_error_weight`` need never exceed the covering radius of the code,

since there are then always lower-weight errors with the same syndrome. If

one sets ``maximum_error_weight`` to a value greater than the covering

radius, then the covering radius will be determined while building the

lookup-table. This lower value is then returned if you query

``decoding_radius`` after construction.

If ``maximum_error_weight`` is left unspecified or set to a number at least

the covering radius of the code, this decoder is complete, i.e. it decodes

every vector in the ambient space.

.. NOTE::

Constructing the lookup table takes time exponential in the length of the

code and the size of the code's base field. Afterwards, the individual

decodings are fast.

INPUT:

- ``code`` -- A code associated to this decoder

- ``maximum_error_weight`` -- (default: ``None``) the maximum number of

errors to look for when building the table. An error is raised if it is

set greater than `n-k`, since this is an upper bound on the covering

radius on any linear code. If ``maximum_error_weight`` is kept

unspecified, it will be set to `n - k`, where `n` is the length of

``code`` and `k` its dimension.

EXAMPLES::

sage: G = Matrix(GF(3), [[1,0,0,1,0,1,0,1,2],[0,1,0,2,2,0,1,1,0],[0,0,1,0,2,2,2,1,2]])

sage: C = LinearCode(G)

sage: D = codes.decoders.LinearCodeSyndromeDecoder(C)

sage: D

Syndrome decoder for [9, 3] linear code over GF(3) handling errors of weight up to 4

If one wants to correct up to a lower number of errors, one can do as follows::

sage: D = codes.decoders.LinearCodeSyndromeDecoder(C, maximum_error_weight=2)

sage: D

Syndrome decoder for [9, 3] linear code over GF(3) handling errors of weight up to 2

If one checks the list of types of this decoder before constructing it,

one will notice it contains the keyword ``dynamic``.

Indeed, the behaviour of the syndrome decoder depends on the maximum

error weight one wants to handle, and how it compares to the minimum

distance and the covering radius of ``code``.

In the following examples, we illustrate this property by computing

different instances of syndrome decoder for the same code.

We choose the following linear code, whose covering radius equals to 4

and minimum distance to 5 (half the minimum distance is 2)::

sage: G = matrix(GF(5), [[1, 0, 0, 0, 0, 4, 3, 0, 3, 1, 0],

....: [0, 1, 0, 0, 0, 3, 2, 2, 3, 2, 1],

....: [0, 0, 1, 0, 0, 1, 3, 0, 1, 4, 1],

....: [0, 0, 0, 1, 0, 3, 4, 2, 2, 3, 3],

....: [0, 0, 0, 0, 1, 4, 2, 3, 2, 2, 1]])

sage: C = LinearCode(G)

In the following examples, we illustrate how the choice of

``maximum_error_weight`` influences the types of the instance of

syndrome decoder, alongside with its decoding radius.

We build a first syndrome decoder, and pick a ``maximum_error_weight``

smaller than both the covering radius and half the minimum distance::

sage: D = C.decoder("Syndrome", maximum_error_weight = 1)

sage: D.decoder_type()

{'always-succeed', 'bounded_distance', 'hard-decision'}

sage: D.decoding_radius()

In that case, we are sure the decoder will always succeed. It is also

a bounded distance decoder.

We now build another syndrome decoder, and this time,

``maximum_error_weight`` is chosen to be bigger than half the minimum distance,

but lower than the covering radius::

sage: D = C.decoder("Syndrome", maximum_error_weight = 3)

sage: D.decoder_type()

{'bounded_distance', 'hard-decision', 'might-error'}

sage: D.decoding_radius()

Here, we still get a bounded distance decoder.

But because we have a maximum error weight bigger than half the

minimum distance, we know it might return a codeword which was not

the original codeword.

And now, we build a third syndrome decoder, whose ``maximum_error_weight``

is bigger than both the covering radius and half the minimum distance::

sage: D = C.decoder("Syndrome", maximum_error_weight = 5)

sage: D.decoder_type()

{'complete', 'hard-decision', 'might-error'}

sage: D.decoding_radius()

In that case, the decoder might still return an unexpected codeword, but

it is now complete. Note the decoding radius is equal to 4: it was

determined while building the syndrome lookup table that any error with

weight more than 4 will be decoded incorrectly. That is because the covering

radius for the code is 4.

The minimum distance and the covering radius are both determined while

computing the syndrome lookup table. They user did not explicitly ask to

compute these on the code ``C``. The dynamic typing of the syndrome decoder

might therefore seem slightly surprising, but in the end is quite

informative.

"""

def __init__(self, code, maximum_error_weight=None):

r"""

TESTS:

If ``maximum_error_weight`` is greater or equal than `n-k`, where `n`

is ``code``'s length, and `k` is ``code``'s dimension,

an error is raised::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: D = codes.decoders.LinearCodeSyndromeDecoder(C, 42)

Traceback (most recent call last):

...

ValueError: maximum_error_weight has to be less than code's length minus its dimension

The Syndrome Decoder of a Hamming code should have types

``minimum-distance`` and ``always-succeed`` (see :trac:`20898`)::

sage: C = codes.HammingCode(GF(5), 3)

sage: D = C.decoder("Syndrome")

sage: C.minimum_distance()

sage: D.maximum_error_weight()

sage: D.decoder_type()

{'always-succeed', 'complete', 'hard-decision', 'minimum-distance'}

"""

n_minus_k = code.length() - code.dimension()

if maximum_error_weight is None:

self._maximum_error_weight = n_minus_k

elif not isinstance(maximum_error_weight, (Integer, int)):

raise ValueError("maximum_error_weight has to be a Sage integer or a Python int")

elif maximum_error_weight > n_minus_k:

raise ValueError("maximum_error_weight has to be less than code's length minus its dimension")

else:

self._maximum_error_weight = maximum_error_weight

super(LinearCodeSyndromeDecoder, self).__init__(code, code.ambient_space(),\

code._default_encoder_name)

self._lookup_table = self._build_lookup_table()

def __eq__(self, other):

r"""

Tests equality between LinearCodeSyndromeDecoder objects.

EXAMPLES::

sage: G = Matrix(GF(3), [[1,0,0,1,0,1,0,1,2],[0,1,0,2,2,0,1,1,0],[0,0,1,0,2,2,2,1,2]])

sage: D1 = codes.decoders.LinearCodeSyndromeDecoder(LinearCode(G))

sage: D2 = codes.decoders.LinearCodeSyndromeDecoder(LinearCode(G))

sage: D1 == D2

True

"""

return isinstance(other, LinearCodeSyndromeDecoder)\

and self.code() == other.code()\

and self.maximum_error_weight() == other.maximum_error_weight()

def _repr_(self):

r"""

Return a string representation of ``self``.

EXAMPLES::

sage: G = Matrix(GF(3), [[1,0,0,1,0,1,0,1,2],[0,1,0,2,2,0,1,1,0],[0,0,1,0,2,2,2,1,2]])

sage: C = LinearCode(G)

sage: D = codes.decoders.LinearCodeSyndromeDecoder(C)

sage: D

Syndrome decoder for [9, 3] linear code over GF(3) handling errors of weight up to 4

"""

return "Syndrome decoder for %s handling errors of weight up to %s" % (self.code(), self.maximum_error_weight())

def _latex_(self):

r"""

Return a latex representation of ``self``.

EXAMPLES::

sage: G = Matrix(GF(3), [[1,0,0,1,0,1,0,1,2],[0,1,0,2,2,0,1,1,0],[0,0,1,0,2,2,2,1,2]])

sage: C = LinearCode(G)

sage: D = codes.decoders.LinearCodeSyndromeDecoder(C)

sage: latex(D)

\textnormal{Syndrome decoder for [9, 3]\textnormal{ Linear code over }\Bold{F}_{3} handling errors of weight up to 4}

"""

return "\\textnormal{Syndrome decoder for %s handling errors of weight up to %s}" % (self.code()._latex_(), self.maximum_error_weight())

@cached_method

def _build_lookup_table(self):

r"""

Builds lookup table for all possible error patterns of weight up to :meth:`maximum_error_weight`.

EXAMPLES::

sage: G = Matrix(GF(3),[

....: [1, 0, 0, 0, 2, 2, 1, 1],

....: [0, 1, 0, 0, 0, 0, 1, 1],

....: [0, 0, 1, 0, 2, 0, 0, 2],

....: [0, 0, 0, 1, 0, 2, 0, 1]])

sage: C = LinearCode(G)

sage: D = codes.decoders.LinearCodeSyndromeDecoder(C, maximum_error_weight = 1)

sage: D._build_lookup_table()

{(0, 0, 0, 0): (0, 0, 0, 0, 0, 0, 0, 0),

(0, 0, 0, 1): (0, 0, 0, 0, 1, 0, 0, 0),

(0, 0, 0, 2): (0, 0, 0, 0, 2, 0, 0, 0),

(0, 0, 1, 0): (0, 0, 1, 0, 0, 0, 0, 0),

(0, 0, 1, 2): (0, 0, 0, 0, 0, 0, 0, 1),

(0, 0, 2, 0): (0, 0, 2, 0, 0, 0, 0, 0),

(0, 0, 2, 1): (0, 0, 0, 0, 0, 0, 0, 2),

(0, 1, 0, 0): (0, 1, 0, 0, 0, 0, 0, 0),

(0, 1, 1, 2): (0, 0, 0, 0, 0, 0, 2, 0),

(0, 2, 0, 0): (0, 2, 0, 0, 0, 0, 0, 0),

(0, 2, 2, 1): (0, 0, 0, 0, 0, 0, 1, 0),

(1, 0, 0, 0): (1, 0, 0, 0, 0, 0, 0, 0),

(1, 2, 0, 2): (0, 0, 0, 0, 0, 1, 0, 0),

(1, 2, 2, 0): (0, 0, 0, 1, 0, 0, 0, 0),

(2, 0, 0, 0): (2, 0, 0, 0, 0, 0, 0, 0),

(2, 1, 0, 1): (0, 0, 0, 0, 0, 2, 0, 0),

(2, 1, 1, 0): (0, 0, 0, 2, 0, 0, 0, 0)}

TESTS:

Check that :trac:`24114` is fixed::

sage: R.<x> = PolynomialRing(GF(3))

sage: f = x^2 + x + 2

sage: K.<a> = f.root_field()

sage: H = Matrix(K,[[1,2,1],[2*a+1,a,1]])

sage: C = codes.from_parity_check_matrix(H)

sage: D = codes.decoders.LinearCodeSyndromeDecoder(C)

sage: D.syndrome_table()

{(0, 0): (0, 0, 0),

(0, 1): (0, 1, 0),

(0, 2): (0, 2, 0),

(0, a): (0, a, 0),

...

(2*a + 2, 2*a): (0, 0, 2),

(2*a + 2, 2*a + 1): (2*a + 2, 2*a + 1, 0),

(2*a + 2, 2*a + 2): (2*a + 2, 2*a + 2, 0)}

"""

t = self._maximum_error_weight

self._code_covering_radius = None

self._code_minimum_distance = None

self._decoder_type = copy(self._decoder_type)

self._decoder_type.remove("dynamic")

C = self.code()

n = C.length()

k = C.dimension()

H = C.parity_check_matrix()

F = C.base_ring()

l = list(F)

zero = F.zero()

#Builds a list of generators of all error positions for all

#possible error weights

if zero in l:

l.remove(zero)

# Remember to include the no-error-vector to handle codes of minimum

# distance 1 gracefully

zero_syndrome = vector(F,[F.zero()]*(n-k))

zero_syndrome.set_immutable()

lookup = { zero_syndrome : vector(F,[F.zero()]*n) }

error_position_tables = [cartesian_product([l]*i) for i in range(1, t+1)]

first_collision = True

#Filling the lookup table

for i in range(1, t+1):

stop = True

patterns = Subsets(range(n), i)

basic = vector(F, n)

for p in patterns:

for error in error_position_tables[i-1]:

ind = 0

e = copy(basic)

for pos in p:

e[pos] = error[ind]

ind += 1

s = H * e

s.set_immutable()

try:

e_cur = lookup[s]

#if this is the first time we see a collision

#we learn the minimum distance of the code

if first_collision:

self._code_minimum_distance = e.hamming_weight() + e_cur.hamming_weight()

first_collision = False

except KeyError:

stop = False

lookup[s] = copy(e)

#if we reached the early termination condition

#we learn the covering radius of the code

if stop:

self._code_covering_radius = i - 1

self._maximum_error_weight = self._code_covering_radius

break

# Update decoder types depending on whether we are decoding up to covering radius

if self._code_covering_radius:

self._decoder_type.add("complete")

else:

self._decoder_type.add("bounded_distance")

# Update decoder types depending on whether we are decoding beyond d/2

if self._code_minimum_distance:

if self._maximum_error_weight == (self._code_minimum_distance-1)//2:

self._decoder_type.update({"minimum-distance","always-succeed"})

else:

# then t > (d-1)/2

self._decoder_type.add("might-error")

else:

self._decoder_type.add("always-succeed")

return lookup

def decode_to_code(self, r):

r"""

Corrects the errors in ``word`` and returns a codeword.

INPUT:

- ``r`` -- a codeword of ``self``

OUTPUT:

- a vector of ``self``'s message space

EXAMPLES::

sage: G = Matrix(GF(3),[

....: [1, 0, 0, 0, 2, 2, 1, 1],

....: [0, 1, 0, 0, 0, 0, 1, 1],

....: [0, 0, 1, 0, 2, 0, 0, 2],

....: [0, 0, 0, 1, 0, 2, 0, 1]])

sage: C = LinearCode(G)

sage: D = codes.decoders.LinearCodeSyndromeDecoder(C, maximum_error_weight = 2)

sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), 2)

sage: c = C.random_element()

sage: r = Chan(c)

sage: c == D.decode_to_code(r)

True

"""

lookup_table = self.syndrome_table()

s = self.code().parity_check_matrix() * r

s.set_immutable()

if s.is_zero():

return r

err = lookup_table[s]

r_corr = copy(r)

for i in range(self.code().length()):

r_corr[i] = r[i] - err[i]

return r_corr

def maximum_error_weight(self):

r"""

Returns the maximal number of errors a received word can have

and for which ``self`` is guaranteed to return a most likely codeword.

Same as ``self.decoding_radius``.

EXAMPLES::

sage: G = Matrix(GF(3), [[1,0,0,1,0,1,0,1,2],[0,1,0,2,2,0,1,1,0],[0,0,1,0,2,2,2,1,2]])

sage: C = LinearCode(G)

sage: D = codes.decoders.LinearCodeSyndromeDecoder(C)

sage: D.maximum_error_weight()

"""

return self._maximum_error_weight

def decoding_radius(self):

r"""

Returns the maximal number of errors a received word can have

and for which ``self`` is guaranteed to return a most likely codeword.

EXAMPLES::

sage: G = Matrix(GF(3), [[1,0,0,1,0,1,0,1,2],[0,1,0,2,2,0,1,1,0],[0,0,1,0,2,2,2,1,2]])

sage: C = LinearCode(G)

sage: D = codes.decoders.LinearCodeSyndromeDecoder(C)

sage: D.decoding_radius()

"""

return self._maximum_error_weight

def syndrome_table(self):

r"""

Return the syndrome lookup table of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: D = codes.decoders.LinearCodeSyndromeDecoder(C)

sage: D.syndrome_table()

{(0, 0, 0): (0, 0, 0, 0, 0, 0, 0),

(0, 0, 1): (0, 0, 0, 1, 0, 0, 0),

(0, 1, 0): (0, 1, 0, 0, 0, 0, 0),

(0, 1, 1): (0, 0, 0, 0, 0, 1, 0),

(1, 0, 0): (1, 0, 0, 0, 0, 0, 0),

(1, 0, 1): (0, 0, 0, 0, 1, 0, 0),

(1, 1, 0): (0, 0, 1, 0, 0, 0, 0),

(1, 1, 1): (0, 0, 0, 0, 0, 0, 1)}

"""

return self._lookup_table

class LinearCodeNearestNeighborDecoder(Decoder):

r"""

Construct a decoder for Linear Codes. This decoder will decode to the

nearest codeword found.

INPUT:

- ``code`` -- A code associated to this decoder

"""

def __init__(self, code):

r"""

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: D = codes.decoders.LinearCodeNearestNeighborDecoder(C)

sage: D

Nearest neighbor decoder for [7, 4] linear code over GF(2)

"""

super(LinearCodeNearestNeighborDecoder, self).__init__(code, code.ambient_space(), \

code._default_encoder_name)

def __eq__(self, other):

r"""

Tests equality between LinearCodeNearestNeighborDecoder objects.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: D1 = codes.decoders.LinearCodeNearestNeighborDecoder(LinearCode(G))

sage: D2 = codes.decoders.LinearCodeNearestNeighborDecoder(LinearCode(G))

sage: D1 == D2

True

"""

return isinstance(other, LinearCodeNearestNeighborDecoder)\

and self.code() == other.code()

def _repr_(self):

r"""

Returns a string representation of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: D = codes.decoders.LinearCodeNearestNeighborDecoder(C)

sage: D

Nearest neighbor decoder for [7, 4] linear code over GF(2)

"""

return "Nearest neighbor decoder for %s" % self.code()

def _latex_(self):

r"""

Returns a latex representation of ``self``.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: D = codes.decoders.LinearCodeNearestNeighborDecoder(C)

sage: latex(D)

\textnormal{Nearest neighbor decoder for }[7, 4]\textnormal{ Linear code over }\Bold{F}_{2}

"""

return "\\textnormal{Nearest neighbor decoder for }%s" % self.code()._latex_()

def decode_to_code(self, r):

r"""

Corrects the errors in ``word`` and returns a codeword.

INPUT:

- ``r`` -- a codeword of ``self``

OUTPUT:

- a vector of ``self``'s message space

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: D = codes.decoders.LinearCodeNearestNeighborDecoder(C)

sage: word = vector(GF(2), (1, 1, 0, 0, 1, 1, 0))

sage: w_err = word + vector(GF(2), (1, 0, 0, 0, 0, 0, 0))

sage: D.decode_to_code(w_err)

(1, 1, 0, 0, 1, 1, 0)

"""

c_min = self.code().zero()

h_min = r.hamming_weight()

for c in self.code():

if (c-r).hamming_weight() < h_min:

h_min = (c-r).hamming_weight()

c_min = c

c_min.set_immutable()

return c_min

def decoding_radius(self):

r"""

Return maximal number of errors ``self`` can decode.

EXAMPLES::

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])

sage: C = LinearCode(G)

sage: D = codes.decoders.LinearCodeNearestNeighborDecoder(C)

sage: D.decoding_radius()

"""

return (self.code().minimum_distance()-1) // 2

####################### registration ###############################

LinearCode._registered_encoders["GeneratorMatrix"] = LinearCodeGeneratorMatrixEncoder

LinearCodeSyndromeDecoder._decoder_type = {"hard-decision", "dynamic"}

LinearCodeNearestNeighborDecoder._decoder_type = {"hard-decision", "always-succeed", "complete"}

Coverage for local/lib/python2.7/site-packages/sage/coding/linear_code.py : 81%

1116 statements 907 run 209 missing 0 excluded