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

return "\\textnormal{{ISD Algorithm ({}) for }}{} \\textnormal{{decoding {} errors}}".format(self._algorithm_name, self.code()._latex_(), _format_decoding_interval(self.decoding_interval()))

class LeeBrickellISDAlgorithm(InformationSetAlgorithm):

r"""

The Lee-Brickell algorithm for information-set decoding.

For a description of the information-set decoding paradigm (ISD), see

:class:`sage.coding.information_set_decoder.LinearCodeInformationSetDecoder`.

This implements the Lee-Brickell variant of ISD, see [LB1988]_ for the

original binary case, and [Pet2010]_ for the `q`-ary extension.

Let `C` be a `[n, k]`-linear code over `GF(q)`, and let `r \in GF(q)^{n}` be

a received word in a transmission. We seek the codeword whose Hamming

distance from `r` is minimal. Let `p` and `w` be integers, such that `0\leq

p\leq w`, Let `G` be a generator matrix of `C`, and for any set of indices

`I`, we write `G_{I}` for the matrix formed by the columns of `G` indexed by

`I`. The Lee-Brickell ISD loops the following until it is successful:

1. Choose an information set `I` of `C`.

2. Compute `r' = r - r_{I}\times G_I^{-1} \times G`

3. Consider every size-`p` subset of `I`, `\{a_1, \dots, a_p\}`.

For each `m = (m_1, \dots, m_p) \in GF(q)^{p}`, compute

the error vector `e = r' - \sum_{i=1}^{p} m_i\times g_{a_i}`,

4. If `e` has a Hamming weight at most `w`, return `r-e`.

INPUT:

- ``code`` -- A linear code for which to decode.

- ``decoding_interval`` -- a pair of integers specifying an interval of

number of errors to correct. Includes both end values.

- ``search_size`` -- (optional) the size of subsets to use on step 3 of the

algorithm as described above. Usually a small number. It has to be at most

the largest allowed number of errors. A good choice will be approximated

if this option is not set; see

:meth:`sage.coding.LeeBrickellISDAlgorithm.calibrate`

for details.

EXAMPLES::

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

sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm

sage: A = LeeBrickellISDAlgorithm(C, (0,4)); A

ISD Algorithm (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding up to 4 errors

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

sage: A = LeeBrickellISDAlgorithm(C, (2,3)); A

ISD Algorithm (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding between 2 and 3 errors

"""

def __init__(self, code, decoding_interval, search_size = None):

r"""

TESTS:

If ``search_size`` is not a positive integer, or is bigger than the

decoding radius, an error will be raised::

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

sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm

sage: LeeBrickellISDAlgorithm(C, (1, 3), search_size=-1)

Traceback (most recent call last):

...

ValueError: The search size parameter has to be a positive integer

sage: LeeBrickellISDAlgorithm(C, (1, 3), search_size=4)

Traceback (most recent call last):

...

ValueError: The search size parameter has to be at most the maximal number of allowed errors

"""

if search_size is not None:

if not isinstance(search_size, (Integer, int)) or search_size < 0:

raise ValueError("The search size parameter has to be a positive integer")

if search_size > decoding_interval[1]:

raise ValueError("The search size parameter has to be at most"

" the maximal number of allowed errors")

super(LeeBrickellISDAlgorithm, self).__init__(code, decoding_interval, "Lee-Brickell",

parameters={ 'search_size': search_size })

self._parameters_specified = True

else:

self._parameters_specified = False

super(LeeBrickellISDAlgorithm, self).__init__(code, decoding_interval, "Lee-Brickell")

def decode(self, r):

r"""

The Lee-Brickell algorithm as described in the class doc.

Note that either parameters must be given at construction time or

:meth:`sage.coding.information_set_decoder.InformationSetAlgorithm.calibrate()`

should be called before calling this method.

INPUT:

- `r` -- a received word, i.e. a vector in the ambient space of

:meth:`decoder.Decoder.code`.

OUTPUT: A codeword whose distance to `r` satisfies ``self.decoding_interval()``.

EXAMPLES::

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

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

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

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

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

sage: C = codes.LinearCode(M)

sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm

sage: A = LeeBrickellISDAlgorithm(C, (2,2))

sage: c = C.random_element()

sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), 2)

sage: r = Chan(c)

sage: c_out = A.decode(r)

sage: (r - c).hamming_weight() == 2

True

"""

import itertools

from sage.all import sample

C = self.code()

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

tau = self.decoding_interval()

p = self.parameters()['search_size']

F = C.base_ring()

G = C.generator_matrix()

Fstar = F.list()[1:]

while True:

# step 1.

I = sample(range(n), k)

Gi = G.matrix_from_columns(I)

try:

Gi_inv = Gi.inverse()

except ZeroDivisionError:

# I was not an information set

continue

Gt = Gi_inv * G

#step 2.

y = r - vector([r[i] for i in I]) * Gt

g = Gt.rows()

#step 3.

for pi in range(p+1):

for A in itertools.combinations(range(k), pi):

for m in itertools.product(Fstar, repeat=pi):

e = y - sum(m[i]*g[A[i]] for i in range(pi))

errs = e.hamming_weight()

if errs >= tau[0] and errs <= tau[1]:

return r - e

def calibrate(self):

r"""

Run some test computations to estimate the optimal search size.

Let `p` be the search size. We should simply choose `p` such that the

average expected time is minimal. The algorithm succeeds when it chooses

an information set with at least `k - p` correct positions, where `k` is

the dimension of the code and `p` the search size. The expected number

of trials we need before this occurs is:

.. MATH::

\binom{n}{k}/(\rho \sum_{i=0}^p \binom{n-\tau}{k-i} \binom{\tau}{i})

Here `\rho` is the fraction of `k` subsets of indices which are

information sets. If `T` is the average time for steps 1 and 2

(including selecting `I` until an information set is found), while `P(i)`

is the time for the body of the ``for``-loop in step 3 for `m` of weight

`i`, then each information set trial takes roughly time `T +

\sum_{i=0}^{p} P(i) \binom{k}{i} (q-1)^i`, where `\GF{q}` is the base

field.

The values `T` and `P` are here estimated by running a few test

computations similar to those done by the decoding algorithm.

We don't explicitly estimate `\rho`.

OUTPUT: Does not output anything but sets private fields used by

:meth:`sage.coding.information_set_decoder.InformationSetAlgorithm.parameters()`

and

:meth:`sage.coding.information_set_decoder.InformationSetAlgorithm.time_estimate()``.

EXAMPLES::

sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm

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

sage: A = LeeBrickellISDAlgorithm(C, (0,3)); A

ISD Algorithm (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding up to 3 errors

sage: A.calibrate()

sage: A.parameters() #random

{'search_size': 1}

sage: A.time_estimate() #random

0.0008162108571427874

If we specify the parameter at construction time, calibrate does not override this choice::

sage: A = LeeBrickellISDAlgorithm(C, (0,3), search_size=2); A

ISD Algorithm (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding up to 3 errors

sage: A.parameters()

{'search_size': 2}

sage: A.calibrate()

sage: A.parameters()

{'search_size': 2}

sage: A.time_estimate() #random

0.0008162108571427874

"""

from sage.all import sample, mean, random_vector, random_matrix, randint

import time

C = self.code()

G = C.generator_matrix()

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

tau = self.decoding_interval()[1]

F = C.base_ring()

q = F.cardinality()

Fstar = F.list()[1:]

def time_information_set_steps():

before = time.clock()

while True:

I = sample(range(n), k)

Gi = G.matrix_from_columns(I)

try:

Gi_inv = Gi.inverse()

except ZeroDivisionError:

continue

return time.clock() - before

def time_search_loop(p):

y = random_vector(F, n)

g = random_matrix(F, p, n).rows()

scalars = [ [ Fstar[randint(0,q-2)] for i in range(p) ]

for s in range(100) ]

before = time.clock()

for m in scalars:

e = y - sum(m[i]*g[i] for i in range(p))

errs = e.hamming_weight()

return (time.clock() - before)/100.

T = mean([ time_information_set_steps() for s in range(5) ])

P = [ time_search_loop(p) for p in range(tau+1) ]

def compute_estimate(p):

iters = 1.* binomial(n, k)/ \

sum( binomial(n-tau, k-i)*binomial(tau,i) for i in range(p+1) )

estimate = iters*(T + \

sum(P[pi] * (q-1)**pi * binomial(k, pi) for pi in range(p+1) ))

return estimate

if self._parameters_specified:

self._time_estimate = compute_estimate(self._parameters['search_size'])

else:

self._calibrate_select([ compute_estimate(p) for p in range(tau+1) ])

def _calibrate_select(self, estimates):

r"""

Internal method used by ``self.calibrate()``.

Given the timing estimates, select the best parameter and set the

appropriate private fields.

INPUT:

- `estimates` - list of time estimates, for the search size set to the

index of the list entry.

OUTPUT: None, but sets the private fields `self._parameters` and

`self._time_estimate`.

TESTS::

sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm

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

sage: A = LeeBrickellISDAlgorithm(C, (0,3)); A

ISD Algorithm (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding up to 3 errors

sage: A._calibrate_select([ 1.0, 2.0, 3.0, 0.5, 0.6, 1.0 ])

sage: A._time_estimate

0.500000000000000

sage: A._parameters

{'search_size': 3}

"""

search_size = 0

for p in range(1, len(estimates)):

if estimates[p] < estimates[search_size]:

search_size = p

self._parameters = { 'search_size': search_size }

self._time_estimate = estimates[search_size]

class LinearCodeInformationSetDecoder(Decoder):

r"""

Information-set decoder for any linear code.

Information-set decoding is a probabilistic decoding strategy that

essentially tries to guess `k` correct positions in the received word,

where `k` is the dimension of the code. A codeword agreeing with the

received word on the guessed position can easily be computed, and their

difference is one possible error vector. A "correct" guess is assumed when

this error vector has low Hamming weight.

The ISD strategy requires choosing how many errors is deemed acceptable. One

choice could be `d/2`, where `d` is the minimum distance of the code, but

sometimes `d` is not known, or sometimes more errors are expected. If one

chooses anything above `d/2`, the algorithm does not guarantee to return a

nearest codeword.

This simple algorithm is not very efficient in itself, but there are numerous

refinements to the strategy. Specifying which strategy to use among those

that Sage knows is done using the ``algorithm`` keyword. If this is not set,

an efficient choice will be made for you.

The various ISD algorithms all need to select a number of parameters. If you

choose a specific algorithm to use, you can pass these parameters as named

parameters directly to this class' constructor. If you don't, efficient

choices will be calibrated for you.

.. WARNING::

If there is no codeword within the specified decoding distance, then the

decoder may never terminate, or it may raise a

:exc:`sage.coding.decoder.DecodingError` exception, depending on the ISD

algorithm used.

INPUT:

- ``code`` -- A linear code for which to decode.

- ``number_errors`` -- an integer, the maximal number of errors to accept as

correct decoding. An interval can also be specified by giving a pair of

integers, where both end values are taken to be in the interval.

- ``algorithm`` -- (optional) the string name of the ISD algorithm to

employ. If this is not set, an appropriate one will be chosen.

A constructed

:class:`sage.coding.information_set_decoder.InformationSetAlgorithm`

object may also be given. In this case ``number_errors`` must match that

of the passed algorithm.

- ``**kwargs`` -- (optional) any number of named arguments passed on to the

ISD algorithm. Such are usually not required, and they can only be set if

``algorithm`` is set to a specific algorithm. See the documentation for

each individual ISD algorithm class for information on any named arguments

they may accept. The easiest way to access this documentation is to first

construct the decoder without passing any named arguments, then accessing

the ISD algorithm using

:meth:`sage.coding.information_set_decoder.LinearCodeInformationSetDecoder.algorithm`,

and then reading the `?` help on the constructed object.

EXAMPLES:

The principal way to access this class is through the

:meth:`sage.code.linear_code.AbstractLinearCode.decoder` method::

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

sage: D = C.decoder("InformationSet", 2); D

Information-set decoder (Lee-Brickell) for [12, 6, 6] Extended Golay code over GF(3) decoding up to 2 errors

You can specify which algorithm you wish to use, and you should do so in

order to pass special parameters to it::

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

sage: D2 = C.decoder("InformationSet", 2, algorithm="Lee-Brickell", search_size=2); D2

Information-set decoder (Lee-Brickell) for [12, 6, 6] Extended Golay code over GF(3) decoding up to 2 errors

sage: D2.algorithm()

ISD Algorithm (Lee-Brickell) for [12, 6, 6] Extended Golay code over GF(3) decoding up to 2 errors

sage: D2.algorithm().parameters()

{'search_size': 2}

If you specify an algorithm which is not known, you get a friendly error message::

sage: C.decoder("InformationSet", 2, algorithm="NoSuchThing")

Traceback (most recent call last):

...

ValueError: Unknown ISD algorithm 'NoSuchThing'. The known algorithms are ['Lee-Brickell'].

You can also construct an ISD algorithm separately and pass that. This is

mostly useful if you write your own ISD algorithms::

sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm

sage: A = LeeBrickellISDAlgorithm(C, (0, 2))

sage: D = C.decoder("InformationSet", 2, algorithm=A); D

Information-set decoder (Lee-Brickell) for [12, 6, 6] Extended Golay code over GF(3) decoding up to 2 errors

When passing an already constructed ISD algorithm, you can't also pass

parameters to the ISD algorithm when constructing the decoder::

sage: C.decoder("InformationSet", 2, algorithm=A, search_size=2)

Traceback (most recent call last):

...

ValueError: ISD algorithm arguments are not allowed when supplying a constructed ISD algorithm

We can also information-set decode non-binary codes::

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

sage: D = C.decoder("InformationSet", 2); D

Information-set decoder (Lee-Brickell) for [12, 6, 6] Extended Golay code over GF(3) decoding up to 2 errors

There are two other ways to access this class::

sage: D = codes.decoders.LinearCodeInformationSetDecoder(C, 2); D

Information-set decoder (Lee-Brickell) for [12, 6, 6] Extended Golay code over GF(3) decoding up to 2 errors

sage: from sage.coding.information_set_decoder import LinearCodeInformationSetDecoder

sage: D = LinearCodeInformationSetDecoder(C, 2); D

Information-set decoder (Lee-Brickell) for [12, 6, 6] Extended Golay code over GF(3) decoding up to 2 errors

"""

def __init__(self, code, number_errors, algorithm=None, **kwargs):

r"""

TESTS:

``number_errors`` has to be either a list of Integers/ints, a tuple of Integers/ints,

or an Integer/int::

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

sage: D = C.decoder("InformationSet", "aa")

Traceback (most recent call last):

...

ValueError: number_errors should be an integer or a pair of integers

If ``number_errors`` is passed as a list/tuple, it has to contain only

two values, the first one being at most the second one::

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

sage: D = C.decoder("InformationSet", (4, 2))

Traceback (most recent call last):

...

ValueError: number_errors should be a positive integer or a valid interval within the positive integers

You cannot ask the decoder to correct more errors than the code length::

sage: D = C.decoder("InformationSet", 25)

Traceback (most recent call last):

...

ValueError: The provided number of errors should be at most the code's length

If ``algorithm`` is not set, additional parameters cannot be passed to

the ISD algorithm::

sage: D = C.decoder("InformationSet", 2, search_size=2)

Traceback (most recent call last):

...

ValueError: Additional arguments to an information-set decoder algorithm are only allowed if a specific algorithm is selected by setting the algorithm keyword

If ``algorithm`` is set to a constructed ISD algorithm, additional

parameters cannot be passed to the ISD algorithm::

sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm

sage: A = LeeBrickellISDAlgorithm(C, (0, 2))

sage: D = C.decoder("InformationSet", 2, A, search_size=3)

Traceback (most recent call last):

...

ValueError: ISD algorithm arguments are not allowed when supplying a constructed ISD algorithm

If ``algorithm`` is set to a constructed

:class:`sage.coding.information_set_decoder.InformationSetAlgorithm`,

then ``number_errors`` must match that of the algorithm::

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

sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm

sage: A = LeeBrickellISDAlgorithm(C, (0, 2))

sage: D = C.decoder("InformationSet", 2, A); D

Information-set decoder (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding up to 2 errors

sage: D = C.decoder("InformationSet", (0,2), A); D

Information-set decoder (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding up to 2 errors

sage: D = C.decoder("InformationSet", 3, A); D

Traceback (most recent call last):

...

ValueError: number_errors must match that of the passed ISD algorithm

"""

if isinstance(number_errors, (Integer, int)):

number_errors = (0, number_errors)

if isinstance(number_errors, (tuple, list)) and len(number_errors) == 2 \

and number_errors[0] in ZZ and number_errors[1] in ZZ:

if 0 > number_errors[0] or number_errors[0] > number_errors[1]:

raise ValueError(

"number_errors should be a positive integer or"

" a valid interval within the positive integers")

if number_errors[1] > code.length():

raise ValueError("The provided number of errors should be at"

" most the code's length")

else:

raise ValueError("number_errors should be an integer or a pair of integers")

self._number_errors = number_errors

super(LinearCodeInformationSetDecoder, self).__init__(

code, code.ambient_space(), code._default_encoder_name)

if algorithm is None:

if kwargs:

raise ValueError("Additional arguments to an information-set decoder"

" algorithm are only allowed if a specific"

" algorithm is selected by setting the algorithm"

" keyword")

algorithm = "Lee-Brickell"

algorithm_names = LinearCodeInformationSetDecoder.known_algorithms(dictionary=True)

if isinstance(algorithm, InformationSetAlgorithm):

if kwargs:

raise ValueError("ISD algorithm arguments are not allowed when"

" supplying a constructed ISD algorithm")

if number_errors != algorithm.decoding_interval():

raise ValueError("number_errors must match that of the passed"

" ISD algorithm")

self._algorithm = algorithm

elif algorithm in algorithm_names:

self._algorithm = algorithm_names[algorithm](code, number_errors, **kwargs)

else:

raise ValueError("Unknown ISD algorithm '{}'."

" The known algorithms are {}."\

.format(algorithm, algorithm_names.keys()))

_known_algorithms = {

"Lee-Brickell": LeeBrickellISDAlgorithm

}

@staticmethod

def known_algorithms(dictionary=False):

r"""

Return the list of ISD algorithms that Sage knows.

Passing any of these to the constructor of

:class:`sage.coding.information_set_decoder.LinearCodeInformationSetDecoder`

will make the ISD decoder use that algorithm.

INPUT:

- ``dictionary`` - optional. If set to ``True``, return a ``dict``

mapping decoding algorithm name to its class.

OUTPUT: a list of strings or a ``dict`` from string to ISD algorithm class.

EXAMPLES::

sage: from sage.coding.information_set_decoder import LinearCodeInformationSetDecoder

sage: sorted(LinearCodeInformationSetDecoder.known_algorithms())

['Lee-Brickell']

"""

if dictionary:

return LinearCodeInformationSetDecoder._known_algorithms

else:

return LinearCodeInformationSetDecoder._known_algorithms.keys()

def algorithm(self):

r"""

Return the ISD algorithm used by this ISD decoder.

EXAMPLES::

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

sage: D = C.decoder("InformationSet", (2,4), "Lee-Brickell")

sage: D.algorithm()

ISD Algorithm (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding between 2 and 4 errors

"""

return self._algorithm

def decode_to_code(self, r):

r"""

Decodes a received word with respect to the associated code of this decoder.

.. WARNING::

If there is no codeword within the decoding radius of this decoder, this

method may never terminate, or it may raise a

:exc:`sage.coding.decoder.DecodingError` exception, depending on the ISD

algorithm used.

INPUT:

- ``r`` -- a vector in the ambient space of :meth:`decoder.Decoder.code`.

OUTPUT: a codeword of :meth:`decoder.Decoder.code`.

EXAMPLES::

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

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

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

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

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

sage: C = LinearCode(M)

sage: c = C.random_element()

sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), 2)

sage: r = Chan(c)

sage: D = C.decoder('InformationSet', 2)

sage: c == D.decode_to_code(r)

True

Information-set decoding a non-binary code::

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

[12, 6, 6] Extended Golay code over GF(3)

sage: c = C.random_element()

sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), 2)

sage: r = Chan(c)

sage: D = C.decoder('InformationSet', 2)

sage: c == D.decode_to_code(r)

True

Let's take a bigger example, for which syndrome decoding or

nearest-neighbor decoding would be infeasible: the `[59, 30]` Quadratic

Residue code over `\GF{3}` has true minimum distance 17, so we can

correct 8 errors::

sage: C = codes.QuadraticResidueCode(59, GF(3))

sage: c = C.random_element()

sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), 2)

sage: r = Chan(c)

sage: D = C.decoder('InformationSet', 8)

sage: c == D.decode_to_code(r) # long time

True

"""

C = self.code()

if r in C:

return r

return self.algorithm().decode(r)

def decoding_radius(self):

r"""

Return the maximal number of errors this decoder can decode.

EXAMPLES::

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

sage: D = C.decoder("InformationSet", 2)

sage: D.decoding_radius()

"""

return self._number_errors[1]

def decoding_interval(self):

r"""

A pair of integers specifying the interval of number of errors this

decoder will attempt to correct.

The interval includes both end values.

EXAMPLES::

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

sage: D = C.decoder("InformationSet", 2)

sage: D.decoding_interval()

(0, 2)

"""

return self._number_errors

def _repr_(self):

r"""

Returns a string representation of this decoding algorithm.

EXAMPLES::

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

sage: D = C.decoder("InformationSet", 2)

sage: D

Information-set decoder (Lee-Brickell) for [24, 12, 8] Extended Golay code over GF(2) decoding up to 2 errors

"""

return "Information-set decoder ({}) for {} decoding {} errors ".format(self.algorithm().name(), self.code(), _format_decoding_interval(self.decoding_interval()))

def _latex_(self):

r"""

Returns a latex representation of this decoding algorithm.

EXAMPLES::

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

sage: from sage.coding.information_set_decoder import LeeBrickellISDAlgorithm

sage: D = C.decoder("InformationSet", 2)

sage: latex(D)

\textnormal{Information-set decoder (Lee-Brickell) for }[24, 12, 8] \textnormal{ Extended Golay Code over } \Bold{F}_{2} \textnormal{decoding up to 2 errors}

"""

return "\\textnormal{{Information-set decoder ({}) for }}{} \\textnormal{{decoding {} errors}}".format(self.algorithm().name(), self.code()._latex_(), _format_decoding_interval(self.decoding_interval()))

LinearCodeInformationSetDecoder._decoder_type = {"hard-decision",

"probabilistic", "not-always-closest", "bounded-distance", "might-fail"}

Coverage for local/lib/python2.7/site-packages/sage/coding/information_set_decoder.py : 98%

178 statements 175 run 3 missing 0 excluded