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r""" 

Canonical forms and automorphism group computation for linear codes over finite fields 

  

We implemented the algorithm described in [Feu2009]_ which computes the unique 

semilinearly isometric code (canonical form) in the equivalence class of a given 

linear code ``C``. Furthermore, this algorithm will return the automorphism 

group of ``C``, too. 

  

The algorithm should be started via a further class 

:class:`~sage.coding.codecan.autgroup_can_label.LinearCodeAutGroupCanLabel`. 

This class removes duplicated columns (up to multiplications 

by units) and zero columns. Hence, we can suppose that the input for the algorithm 

developed here is a set of points in `PG(k-1, q)`. 

  

The implementation is based on the class 

:class:`sage.groups.perm_gps.partn_ref2.refinement_generic.PartitionRefinement_generic`. 

See the description of this algorithm in 

:mod:`sage.groups.perm_gps.partn_ref2.refinement_generic`. 

In the language given there, we have to implement the group action of 

`G = (GL(k,q) \times {\GF{q}^*}^n ) \rtimes Aut(\GF{q})` on the set `X = 

(\GF{q}^k)^n` of `k \times n` matrices over `\GF{q}` (with the above 

restrictions). 

  

The derived class here implements the stabilizers 

`G_{\Pi^{(I)}(x)}` of the projections `\Pi^{(I)}(x)` of `x` to 

the coordinates specified in the sequence `I`. Furthermore, we implement 

the inner minimization, i.e. the computation of a canonical form of 

the projection `\Pi^{(I)}(x)` under the action of `G_{\Pi^{(I^{(i-1)})}(x)}` . 

Finally, we provide suitable homomorphisms of group actions for the refinements 

and methods to compute the applied group elements in `G \rtimes S_n`. 

  

The algorithm also uses Jeffrey Leon's idea of maintaining an 

invariant set of codewords which is computed in the beginning, see 

:meth:`~sage.coding.codecan.codecan.PartitionRefinementLinearCode._init_point_hyperplane_incidence`. 

An example for such a set is the set of all codewords of weight `\leq w` for 

some uniquely defined `w`. In our case, we interpret the codewords as a set of 

hyperplanes (via the corresponding information word) and compute invariants of 

the bipartite, colored derived subgraph of the point-hyperplane incidence graph, 

see :meth:`PartitionRefinementLinearCode._point_refine` and 

:meth:`PartitionRefinementLinearCode._hyp_refine`. 

  

Since we are interested in subspaces (linear codes) instead of matrices, our 

group elements returned in 

:meth:`PartitionRefinementLinearCode.get_transporter` and 

:meth:`PartitionRefinementLinearCode.get_autom_gens` 

will be elements in the group 

`({\GF{q}^*}^n \rtimes Aut(\GF{q})) \rtimes S_n = 

({\GF{q}^*}^n \rtimes (Aut(\GF{q}) \times S_n)`. 

  

AUTHORS: 

  

- Thomas Feulner (2012-11-15): initial version 

  

REFERENCES: 

  

- [Feu2009] 

  

EXAMPLES: 

  

Get the canonical form of the Simplex code:: 

  

sage: from sage.coding.codecan.codecan import PartitionRefinementLinearCode 

sage: mat = codes.HammingCode(GF(3), 3).dual_code().generator_matrix() 

sage: P = PartitionRefinementLinearCode(mat.ncols(), mat) 

sage: cf = P.get_canonical_form(); cf 

[1 0 0 0 0 1 1 1 1 1 1 1 1] 

[0 1 0 1 1 0 0 1 1 2 2 1 2] 

[0 0 1 1 2 1 2 1 2 1 2 0 0] 

  

The transporter element is a group element which maps the input 

to its canonical form:: 

  

sage: cf.echelon_form() == (P.get_transporter() * mat).echelon_form() 

True 

  

The automorphism group of the input, i.e. the stabilizer under this group action, 

is returned by generators:: 

  

sage: P.get_autom_order_permutation() == GL(3, GF(3)).order()/(len(GF(3))-1) 

True 

sage: A = P.get_autom_gens() 

sage: all( [(a*mat).echelon_form() == mat.echelon_form() for a in A]) 

True 

  

REFERENCES: 

  

.. [Feu2009] Thomas Feulner, The Automorphism Groups of Linear Codes and 

Canonical Representatives of Their Semilinear Isometry Classes, Advances in 

Mathematics of Communications 3 (4), pp. 363-383, 2009. 

  

""" 

  

#******************************************************************************* 

# Copyright (C) 2012 Thomas Feulner <thomas.feulner@uni-bayreuth.de> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# as published by the Free Software Foundation; either version 2 of 

# the License, or (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#******************************************************************************* 

from __future__ import absolute_import 

  

from copy import copy 

from cysignals.memory cimport check_allocarray, sig_free 

  

from sage.rings.integer cimport Integer 

from sage.matrix.matrix cimport Matrix 

from sage.groups.perm_gps.permgroup import PermutationGroup 

cimport sage.groups.perm_gps.partn_ref2.refinement_generic 

from sage.modules.finite_submodule_iter cimport FiniteFieldsubspace_projPoint_iterator as FFSS_projPoint 

from sage.groups.perm_gps.partn_ref.data_structures cimport * 

include "sage/data_structures/bitset.pxi" 

  

  

cdef class InnerGroup: 

r""" 

This class implements the stabilizers `G_{\Pi^{(I)}(x)}` described in 

:mod:`sage.groups.perm_gps.partn_ref2.refinement_generic` with 

`G = (GL(k,q) \times \GF{q}^n ) \rtimes Aut(\GF{q})`. 

  

Those stabilizers can be stored as triples: 

  

- ``rank`` - an integer in `\{0, \ldots, k\}` 

- ``row_partition`` - a partition of `\{0, \ldots, k-1\}` with 

discrete cells for all integers `i \geq rank`. 

- ``frob_pow`` an integer in `\{0, \ldots, r-1\}` if `q = p^r` 

  

The group `G_{\Pi^{(I)}(x)}` contains all elements `(A, \varphi, \alpha) \in G`, 

where 

  

- `A` is a `2 \times 2` blockmatrix, whose upper left matrix 

is a `k \times k` diagonal matrix whose entries `A_{i,i}` are constant 

on the cells of the partition ``row_partition``. 

The lower left matrix is zero. 

And the right part is arbitrary. 

- The support of the columns given by `i \in I` intersect exactly one 

cell of the partition. The entry `\varphi_i` is equal to the entries 

of the corresponding diagonal entry of `A`. 

- `\alpha` is a power of `\tau^{frob_pow}`, where `\tau` denotes the 

Frobenius automorphism of the finite field `\GF{q}`. 

  

See [Feu2009]_ for more details. 

""" 

def __cinit__(self, k=0, algorithm="semilinear", **kwds): 

r""" 

See :class:`sage.coding.codecan.codecan.InnerGroup` 

  

INPUT: 

  

- ``k`` -- an integer, gives the dimension of the matrix component 

- ``algorithm`` -- either 

  

* "semilinear" -- full group 

* "linear" -- no field automorphisms, i.e. `G = (GL(k,q) \times \GF{q}^n )` 

* "permutational -- no field automorphisms and no column multiplications 

  

i.e. `G = GL(k,q)` 

- ``transporter`` (optional) -- set to an element of the group 

:class:`sage.groups.semimonomial_transformations.semimonomial_transformation_group.SemimonomialTransformationGroup` 

if you would like to modify this element simultaneously 

  

EXAMPLES:: 

  

sage: from sage.coding.codecan.codecan import InnerGroup 

sage: IG = InnerGroup(10) 

sage: IG = InnerGroup(10, "linear") 

sage: IG = InnerGroup(10, "permutational") 

  

:: 

  

sage: S = SemimonomialTransformationGroup(GF(4, 'a'), 8) 

sage: IG = InnerGroup(3, transporter=S.an_element()) 

""" 

  

self.rank = 0 

if k > 0: 

self.row_partition = OP_new(k) 

  

if algorithm == "permutational": 

self.frob_pow = 0 

self.permutational_only = 1 

for i in range(1, k): 

OP_join(self.row_partition, 0, i) 

else: 

self.permutational_only = 0 

if algorithm == "semilinear": 

self.frob_pow = 1 

elif algorithm == "linear": 

self.frob_pow = 0 

  

  

self.compute_transporter = False 

if "transporter" in kwds: 

self.transporter = kwds["transporter"] 

self.compute_transporter = True 

  

def __dealloc__(self): 

r""" 

Deallocates ``self``. 

""" 

OP_dealloc(self.row_partition) 

  

cdef int get_rep(self, int pos): 

""" 

Get the index of the cell of ``self.row_partition`` containing ``pos``. 

""" 

return OP_find(self.row_partition, pos) 

  

cdef bint has_semilinear_action(self): 

""" 

Returns ``True`` iff the field automorphism group component of ``self`` 

is non-trivial. 

""" 

return (self.frob_pow > 0) 

  

cdef int join_rows(self, int rep1, int rep2): 

""" 

Join the cells with unique representatives 

``rep1`` and ``rep2`` of ``self.row_partition``. 

Return the index of the join. 

""" 

OP_join(self.row_partition, rep1, rep2) 

return self.get_rep(rep1) 

  

cdef void copy_from(self, InnerGroup other): 

""" 

Copy the group ``other`` to ``self``. 

""" 

self.rank = other.rank 

self.frob_pow = other.frob_pow 

self.permutational_only = other.permutational_only 

OP_copy_from_to(other.row_partition, self.row_partition) 

  

cdef minimize_by_row_mult(self, FreeModuleElement w): 

r""" 

We suppose `v \in \GF{q}^k` and the entries `v_i = 0` for all 

``i >= self.rank``. 

We compute the smallest element ``w`` in the orbit of ``v`` under the 

group action of the matrix components of all elements in ``self``. 

  

We return ``d, w``, where ``d`` is a dictionary mapping 

``self.row_partition`` (accessed via their unique representatives) 

to its necessary multiplication. Non-occurring cells are multiplicated 

by 1. 

""" 

cdef FreeModuleElement v = w.__copy__() 

cdef dict d = dict() 

if self.permutational_only: 

return d, v 

  

cdef list nz_pos = v.nonzero_positions() 

for r in nz_pos: 

r_rep = self.get_rep(r) 

if r_rep not in d: 

d[r_rep] = v[r] ** (-1) 

v[r] = 1 

else: 

v[r] *= d[r_rep] 

return d, v 

  

cdef minimize_matrix_col(self, object m, int pos, list fixed_minimized_cols, 

bint *group_changed): 

r""" 

Minimize the column at position ``pos`` of the matrix ``m`` by the 

action of ``self``. ``m`` should have no zero column. ``self`` is set to 

the stabilizer of this column. 

  

We return ``group_changed, mm`` where ``group_changed`` is a boolean 

indicating if ``self`` got changed and ``mm`` is the modification of 

``m``. 

  

In the array ``fixed_minimized_cols`` we store, those 

columns of ``m`` which are known to be invariant under ``self``. 

""" 

group_changed[0] = False 

cdef SemimonomialTransformation my_trans 

cdef FreeModuleElement act_col = m.column(pos) 

cdef int pivot = -1 

cdef list nz_pos = act_col.nonzero_positions() 

cdef int applied_frob, i, col, row, first_nz_rep 

  

F = m.base_ring() 

  

for i in nz_pos: 

if i >= self.rank: 

pivot = i 

break 

  

if pivot == -1: 

if self.permutational_only: 

return m 

# this column is linearly dependent on those already fixed 

first_nz = nz_pos.pop(0) 

first_nz_rep = self.get_rep(first_nz) 

factor = m[first_nz, pos] ** (-1) 

m.rescale_col(pos, factor) 

  

if self.compute_transporter: 

n = self.transporter.parent().degree() 

v = (F.one(),)*(pos) + (factor**(-1), ) + (F.one(),)*(n-pos-1) 

my_trans = self.transporter.parent()(v=v) 

  

d, _ = self.minimize_by_row_mult(factor * act_col) 

d.pop(first_nz_rep) 

if len(d): # there is at least one more multiplication 

group_changed[0] = True 

for i in range(self.rank): 

factor = d.get(self.get_rep(i)) 

if factor and not factor.is_zero(): 

m.rescale_row(i, factor) 

for i in d: 

first_nz_rep = self.join_rows(first_nz_rep, i) 

# rescale the already fixed part by column multiplications 

for col in fixed_minimized_cols: 

col_nz = m.column(col).nonzero_positions() 

if len(col_nz) > 0: 

row = col_nz[0] 

if self.compute_transporter: 

my_trans.v = (my_trans.v[:col] + (m[row, col],) + 

my_trans.v[col+1:]) 

m.rescale_col(col, m[row, col] ** (-1)) 

if self.has_semilinear_action(): 

applied_frob = 0 

self.minimize_by_frobenius(m[nz_pos].column(pos), &applied_frob, &self.frob_pow) 

f = F.hom([F.gen() ** (F.characteristic() ** applied_frob)]) 

m = m.apply_map(f) # this would change the reference! 

  

if self.compute_transporter: 

my_trans.v = tuple([my_trans.v[i].frobenius(applied_frob) 

for i in range(len(my_trans.v))]) 

my_trans.alpha = f 

  

if self.compute_transporter: 

self.transporter = my_trans * self.transporter 

else: 

# this column is linearly independent on those already fixed, 

# map it to the self._rank-th unit vector 

group_changed[0] = True 

self.gaussian_elimination(m, pos, pivot, nz_pos) 

self.rank += 1 

return m 

  

cdef void gaussian_elimination(self, object m, int pos, int pivot, list nz_pos): 

r""" 

Minimize the column at position ``pos`` of the matrix ``m`` by the 

action of ``self``. We know that there is some nonzero entry of this 

column at ``pivot >= self.rank``. All nonzero entries are stored in 

the list ``nz_pos``. 

  

``self`` is not modified by this function, but ``m`` is. 

""" 

nz_pos.remove(pivot) 

m.rescale_row(pivot, m[pivot, pos] ** (-1)) 

  

for r in nz_pos: 

m.add_multiple_of_row(r, pivot, -m[r, pos]) # Gaussian elimination 

if pivot != self.rank: 

m.swap_rows(self.rank, pivot) 

  

cdef InnerGroup _new_c(self): 

r""" 

Make a new copy of ``self``. 

""" 

cdef InnerGroup res = InnerGroup() 

res.frob_pow = self.frob_pow 

res.rank = self.rank 

res.row_partition = OP_copy(self.row_partition) 

res.permutational_only = self.permutational_only 

return res 

  

cdef SemimonomialTransformation get_transporter(self): 

r""" 

Return the group element we have applied. Should only be called if 

we passed an element in 

:meth:`sage.coding.codecan.codecan.InnerGroup.__cinit__`. 

""" 

return self.transporter 

  

def __repr__(self): 

""" 

EXAMPLES:: 

  

sage: from sage.coding.codecan.codecan import InnerGroup 

sage: InnerGroup(10) 

Subgroup of (GL(k,q) times \GF{q}^n ) rtimes Aut(\GF{q}) with rank = 0, 

frobenius power = 1 and partition = 0 -> 0 1 -> 1 2 -> 2 3 -> 3 4 -> 4 5 -> 5 

6 -> 6 7 -> 7 8 -> 8 9 -> 9 

""" 

return "Subgroup of (GL(k,q) times \GF{q}^n ) rtimes Aut(\GF{q}) " + \ 

"with rank = %s, frobenius power = %s and partition =%s" % (self.rank, 

self.frob_pow, OP_string(self.row_partition)) 

  

cdef void minimize_by_frobenius(self, object v, int *applied_frob, int *stab_pow): 

r""" 

Minimize the vector ``v \in \GF{q}^k`` by the 

action of the field automorphism component of ``self``. 

``self`` and ``v`` are not modified by this function. 

  

Let `\tau` denote the Frobenius automorphism of ``\GF{q}``. Then 

``applied_frob``-th power of `\tau` will give us the minimal element. 

The ``stab_pow``-th power of `\tau` will generate the stabilizer of `v`. 

""" 

stab_pow[0] = self.frob_pow 

applied_frob[0] = 0 

cdef int loc_frob, min_pow = 0 

for el in v: 

x = el.frobenius(applied_frob[0]) 

y = x # the elements in the cyclic(!) orbit 

m = x # a candidate for the minimal element 

  

loc_frob = 0 

min_pow = 0 

  

while True: 

loc_frob += 1 

y = y.frobenius(stab_pow[0]) 

if y == x: 

break 

if y < m: 

m = y 

min_pow = loc_frob 

  

# now x.frobenius(stab_pow*loc_frob) == x 

applied_frob[0] += min_pow * stab_pow[0] 

stab_pow[0] *= loc_frob 

if stab_pow[0] == el.parent().degree(): 

stab_pow[0] = 0 

break # for 

  

cpdef int get_frob_pow(self): 

r""" 

Return the power of the Frobenius automorphism which generates 

the corresponding component of ``self``. 

  

EXAMPLES:: 

  

sage: from sage.coding.codecan.codecan import InnerGroup 

sage: I = InnerGroup(10) 

sage: I.get_frob_pow() 

1 

""" 

return self.frob_pow 

  

cpdef column_blocks(self, mat): 

r""" 

Let ``mat`` be a matrix which is stabilized by ``self`` having no zero 

columns. We know that for each column of ``mat`` there is a uniquely 

defined cell in ``self.row_partition`` having a nontrivial intersection 

with the support of this particular column. 

  

This function returns a partition (as list of lists) of the columns 

indices according to the partition of the rows given by ``self``. 

  

EXAMPLES:: 

  

sage: from sage.coding.codecan.codecan import InnerGroup 

sage: I = InnerGroup(3) 

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

sage: I.column_blocks(mat) 

[[1], [0], [2]] 

""" 

if self.row_partition.num_cells == 1: 

return [range(mat.ncols())] 

  

r = [[] for i in range(mat.ncols()) ] 

cols = iter(mat.columns()) 

for i in range(mat.ncols()): 

# there should be no zero columns by assumption! 

m = OP_find(self.row_partition, next(cols).nonzero_positions()[0]) 

r[m].append(i) 

return [ x for x in r if len(x) > 0 ] 

  

cdef class PartitionRefinementLinearCode(PartitionRefinement_generic): 

""" 

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

  

EXAMPLES:: 

  

sage: from sage.coding.codecan.codecan import PartitionRefinementLinearCode 

sage: mat = codes.HammingCode(GF(3), 3).dual_code().generator_matrix() 

sage: P = PartitionRefinementLinearCode(mat.ncols(), mat) 

sage: cf = P.get_canonical_form(); cf 

[1 0 0 0 0 1 1 1 1 1 1 1 1] 

[0 1 0 1 1 0 0 1 1 2 2 1 2] 

[0 0 1 1 2 1 2 1 2 1 2 0 0] 

  

:: 

  

sage: cf.echelon_form() == (P.get_transporter() * mat).echelon_form() 

True 

  

:: 

  

sage: P.get_autom_order_permutation() == GL(3, GF(3)).order()/(len(GF(3))-1) 

True 

sage: A = P.get_autom_gens() 

sage: all( [(a*mat).echelon_form() == mat.echelon_form() for a in A]) 

True 

""" 

def __cinit__(self): 

r""" 

Initialization. See :meth:`__init__`. 

  

TESTS:: 

  

sage: from sage.coding.codecan.codecan import PartitionRefinementLinearCode 

sage: C = PartitionRefinementLinearCode.__new__(PartitionRefinementLinearCode, 0) 

""" 

self._hyp2points = NULL 

self._points2hyp = NULL 

self._hyp_part = NULL 

self._hyp_refine_vals_scratch = NULL 

self._nr_of_supp_refine_calls = 0 

self._nr_of_point_refine_calls = 0 

self._stored_states = dict() 

  

def __init__(self, n, generator_matrix, P=None, algorithm_type="semilinear"): 

r""" 

Initialization, we immediately start the algorithm 

(see :mod:`sage.coding.codecan.codecan`) 

to compute the canonical form and automorphism group of the linear code 

generated by ``generator_matrix``. 

  

INPUT: 

  

- ``n`` -- an integer 

- ``generator_matrix`` -- a `k \times n` matrix over `\GF{q}` of full row rank, 

i.e. `k<n` and without zero columns. 

- partition (optional) -- a partition (as list of lists) of the set 

`\{0, \ldots, n-1\}` which restricts the action of the permutational 

part of the group to the stabilizer of this partition 

- algorithm_type (optional) -- use one of the following options 

  

* "semilinear" - full group 

* "linear" - no field automorphisms, i.e. `G = (GL(k,q) \times \GF{q}^n )` 

* "permutational - no field automorphisms and no column multiplications 

i.e. `G = GL(k,q)` 

  

EXAMPLES:: 

  

sage: from sage.coding.codecan.codecan import PartitionRefinementLinearCode 

sage: mat = codes.HammingCode(GF(3), 3).dual_code().generator_matrix() 

sage: P = PartitionRefinementLinearCode(mat.ncols(), mat) 

  

:: 

  

sage: from sage.coding.codecan.codecan import PartitionRefinementLinearCode 

sage: mat = codes.HammingCode(GF(3), 3).dual_code().generator_matrix() 

sage: P = PartitionRefinementLinearCode(mat.ncols(), mat) 

""" 

self._k = generator_matrix.nrows() 

self._q = len(generator_matrix.base_ring()) 

self._matrix = copy(generator_matrix) 

self._root_matrix = generator_matrix 

self._supp_refine_vals = _BestValStore(n) 

self._point_refine_vals = _BestValStore(n) 

# self._hyp_refine_vals will initialized after 

# we computed the set of codewords 

  

self._run(P, algorithm_type) 

  

def __dealloc__(self): 

r""" 

Deallocates ``self``. 

""" 

cdef int i 

if self._points2hyp is not NULL: 

for i in range(self._n): 

bitset_free(self._points2hyp[i]) 

sig_free(self._points2hyp) 

  

if self._points2hyp is not NULL: 

for i in range(self._hyp_part.degree): 

bitset_free(self._hyp2points[i]) 

sig_free(self._hyp2points) 

  

PS_dealloc(self._hyp_part) 

sig_free(self._hyp_refine_vals_scratch) 

  

def __repr__(self): 

""" 

EXAMPLES:: 

  

sage: from sage.coding.codecan.codecan import PartitionRefinementLinearCode 

sage: mat = codes.HammingCode(GF(3), 3).dual_code().generator_matrix() 

sage: PartitionRefinementLinearCode(mat.ncols(), mat) 

Canonical form algorithm for linear code generated by 

[1 0 1 1 0 1 0 1 1 1 0 1 1] 

[0 1 1 2 0 0 1 1 2 0 1 1 2] 

[0 0 0 0 1 1 1 1 1 2 2 2 2] 

""" 

return "Canonical form algorithm for linear code generated" + \ 

" by\n%s" % (self._root_matrix) 

  

def _run(self, P, algorithm_type): 

""" 

Start the main algorithm, this method is called in :meth:`init`. 

See this method for the description of the input. 

  

EXAMPLES:: 

  

sage: from sage.coding.codecan.codecan import PartitionRefinementLinearCode 

sage: mat = codes.HammingCode(GF(3), 3).dual_code().generator_matrix() 

sage: P = PartitionRefinementLinearCode(mat.ncols(), mat) #indirect doctest 

sage: P.get_canonical_form() 

[1 0 0 0 0 1 1 1 1 1 1 1 1] 

[0 1 0 1 1 0 0 1 1 2 2 1 2] 

[0 0 1 1 2 1 2 1 2 1 2 0 0] 

""" 

self._init_point_hyperplane_incidence() 

F = self._matrix.base_ring() 

if F.order() == 2: 

algorithm_type = "permutational" 

elif self._matrix.base_ring().is_prime_field() and algorithm_type != "permutational": 

algorithm_type = "linear" 

self._inner_group = InnerGroup(self._k, algorithm_type) 

  

self._init_partition_stack(P) 

self._init_point_hyperplane_incidence() 

self._start_Sn_backtrack() #start the main computation 

  

# up to now, we just computed the permutational part of the group action 

# compute the other components of the transporter 

from sage.combinat.permutation import Permutation 

from sage.groups.semimonomial_transformations.semimonomial_transformation_group import SemimonomialTransformationGroup 

from sage.groups.perm_gps.permgroup_named import SymmetricGroup 

  

S = SemimonomialTransformationGroup(self._matrix.base_ring(), self._n) 

S_n = SymmetricGroup(self._n) 

  

self._transporter = S(perm= S_n(self._to_best.sage())) 

self._transporter, self._best_candidate, remaining_inner_group = self._compute_group_element(self._transporter, algorithm_type) 

  

# compute the other components of the automorphism group generators 

self._autom_group_generators = [] 

transp_inv = self._transporter ** (-1) 

  

for a in self._known_automorphisms.small_generating_set(): 

x = S(perm=self._transporter.get_perm() * Permutation(S_n(a))) 

x, _, _ = self._compute_group_element(x, algorithm_type) 

self._autom_group_generators.append(transp_inv * x) 

  

if algorithm_type == "permutational": 

self._inner_group_stabilizer_order = 1 

else: 

P = remaining_inner_group.column_blocks(self._best_candidate) 

for p in P: 

x = S(v=[ F.primitive_element() if i in p else F.one() for i in range(self._n) ]) 

self._autom_group_generators.append(transp_inv * x * self._transporter) 

self._inner_group_stabilizer_order = (len(F) - 1) ** len(P) 

  

  

if remaining_inner_group.get_frob_pow() > 0: 

x = S(autom=F.hom([F.primitive_element() ** (remaining_inner_group.get_frob_pow() * F.characteristic())])) 

self._autom_group_generators.append(transp_inv * x * self._transporter) 

self._inner_group_stabilizer_order *= Integer(F.degree() / remaining_inner_group.get_frob_pow()) 

  

cdef _compute_group_element(self, SemimonomialTransformation trans, str algorithm_type): 

""" 

Apply ``trans`` to ``self._root_matrix`` and minimize the this matrix 

column by column under the inner minimization. The action is 

simoultaneously applied to ``trans``. 

  

The output of this function is a triple containing, the modified 

group element ``trans``, the minimized matrix and the stabilizer of this 

matrix under the inner group. 

""" 

cdef InnerGroup inner_group = InnerGroup(self._k, algorithm_type, transporter=trans) 

cdef bint group_changed = False 

cdef int i 

cdef list fixed_pos = [] 

mat = trans * self._root_matrix 

  

for i in range(self._n): 

mat = inner_group.minimize_matrix_col(mat, i, fixed_pos, 

&group_changed) 

fixed_pos.append(i) 

  

trans = inner_group.get_transporter() 

return trans, mat, inner_group 

  

def get_canonical_form(self): 

r""" 

Return the canonical form for this matrix. 

  

EXAMPLES:: 

  

sage: from sage.coding.codecan.codecan import PartitionRefinementLinearCode 

sage: mat = codes.HammingCode(GF(3), 3).dual_code().generator_matrix() 

sage: P1 = PartitionRefinementLinearCode(mat.ncols(), mat) 

sage: CF1 = P1.get_canonical_form() 

sage: s = SemimonomialTransformationGroup(GF(3), mat.ncols()).an_element() 

sage: P2 = PartitionRefinementLinearCode(mat.ncols(), s*mat) 

sage: CF1 == P2.get_canonical_form() 

True 

""" 

return self._best_candidate 

  

def get_transporter(self): 

""" 

Return the transporter element, mapping the initial matrix to its 

canonical form. 

  

EXAMPLES:: 

  

sage: from sage.coding.codecan.codecan import PartitionRefinementLinearCode 

sage: mat = codes.HammingCode(GF(3), 3).dual_code().generator_matrix() 

sage: P = PartitionRefinementLinearCode(mat.ncols(), mat) 

sage: CF = P.get_canonical_form() 

sage: t = P.get_transporter() 

sage: (t*mat).echelon_form() == CF.echelon_form() 

True 

""" 

return self._transporter 

  

def get_autom_gens(self): 

""" 

Return generators of the automorphism group of the initial matrix. 

  

EXAMPLES:: 

  

sage: from sage.coding.codecan.codecan import PartitionRefinementLinearCode 

sage: mat = codes.HammingCode(GF(3), 3).dual_code().generator_matrix() 

sage: P = PartitionRefinementLinearCode(mat.ncols(), mat) 

sage: A = P.get_autom_gens() 

sage: all( [(a*mat).echelon_form() == mat.echelon_form() for a in A]) 

True 

""" 

return self._autom_group_generators 

  

def get_autom_order_inner_stabilizer(self): 

""" 

Return the order of the stabilizer of the initial matrix under 

the action of the inner group `G`. 

  

EXAMPLES:: 

  

sage: from sage.coding.codecan.codecan import PartitionRefinementLinearCode 

sage: mat = codes.HammingCode(GF(3), 3).dual_code().generator_matrix() 

sage: P = PartitionRefinementLinearCode(mat.ncols(), mat) 

sage: P.get_autom_order_inner_stabilizer() 

2 

sage: mat2 = Matrix(GF(4, 'a'), [[1,0,1], [0,1,1]]) 

sage: P2 = PartitionRefinementLinearCode(mat2.ncols(), mat2) 

sage: P2.get_autom_order_inner_stabilizer() 

6 

""" 

return self._inner_group_stabilizer_order 

  

cdef _init_point_hyperplane_incidence(self): 

""" 

Compute a set of codewords `W` of `C` (generated by self) which is compatible 

with the group action, i.e. if we start with some other code `(g,\pi)C` 

the result should be `(g,\pi)W`. 

  

The set `W` will consist of all normalized codewords up to some weight 

`w`, where `w` is the smallest integer such that `W` spans the linear code `C`. 

  

This set is then transformed to an incidence matrix ``self._points2hyp`` 

of the point-hyperplane graph (points correspond to rows, hyperplanes to 

columns). The hyperplanes correspond to the information 

words. For performance reasons, we also store the transpose 

``self._hyp2points`` of ``self._points2hyp``. 

  

This graph will be later used in the refinement procedures. 

""" 

from sage.matrix.constructor import matrix 

cdef FFSS_projPoint iter = FFSS_projPoint(self._matrix) 

  

ambient_space = (self._matrix.base_ring()) ** (self._n) 

weights2size = [0] * (self.len() + 1) 

W = [[] for xx in range(self.len() + 1)] 

span = [ambient_space.zero_subspace()] * (self.len() + 1) 

min_weight = self.len() 

max_weight = self.len() 

  

while True: # compute an invariant set of (normalized) codewords which span the subspace 

try: 

cw = next(iter) 

except StopIteration: 

break 

w = cw.hamming_weight() 

if min_weight > w: 

min_weight = w 

if w <= max_weight: 

X = ambient_space.subspace([cw]) 

for i in range(w, max_weight): 

old_dim = span[i].dimension() 

span[i] += X 

if span[i].dimension() == old_dim: 

break # this will also be the case for all others 

if old_dim + 1 == self._k: 

# the codewords of weight <= max_weight span the code 

max_weight = i 

break 

W[w].append(cw) 

  

flat_W = sum(W[min_weight: max_weight + 1], []) 

cdef int __hyp2points_size = len(flat_W) 

self._hyp_part = PS_new(__hyp2points_size, 1) 

s = -1 

for x in W[min_weight: max_weight]: 

s += len(x) 

if s >= 0: 

self._hyp_part.levels[s] = 0 

  

self._points2hyp = <bitset_t*>check_calloc(self._n, sizeof(bitset_t)) 

for i in range(self._n): 

bitset_init(self._points2hyp[i], self._hyp_part.degree) 

  

self._hyp2points = <bitset_t*>check_calloc(self._hyp_part.degree, sizeof(bitset_t)) 

for i in range(self._hyp_part.degree): 

bitset_init(self._hyp2points[i], self._n) 

for j in flat_W[i].support(): 

bitset_add(self._hyp2points[i], j) 

bitset_add(self._points2hyp[j], i) 

  

self._hyp_refine_vals_scratch = <long*>check_allocarray( 

self._hyp_part.degree, sizeof(long)) 

  

self._hyp_refine_vals = _BestValStore(self._hyp_part.degree) 

  

cdef bint _minimization_allowed_on_col(self, int pos): 

r""" 

Decide if we are allowed to perform the inner minimization on position 

``pos`` which is supposed to be a singleton. For linear codes over finite 

fields, we can always return ``True``. 

""" 

return True 

  

cdef bint _inner_min_(self, int pos, bint *inner_group_changed): 

r""" 

Minimize the node by the action of the inner group on the ``pos``-th position. 

Sets ``inner_group_changed`` to ``True`` if and only if the inner group 

has changed. 

  

INPUT: 

  

- ``pos`` -- A position in ``range(self.n)`` 

  

OUTPUT: 

  

- ``True`` if and only if the actual node compares less or equal 

to the candidate for the canonical form. 

""" 

self._matrix = self._inner_group.minimize_matrix_col(self._matrix, pos, 

self._fixed_minimized, inner_group_changed) 

  

# finally compare the new column with the best candidate 

if self._is_candidate_initialized: 

A = self._matrix.column(pos) 

B = self._best_candidate.column( 

self._inner_min_order_best[len(self._fixed_minimized)]) 

if B < A: 

return False 

if A < B: 

# the next leaf will become the next candidate 

self._is_candidate_initialized = False 

return True 

  

cdef bint _refine(self, bint *part_changed, 

bint inner_group_changed, bint first_step): 

""" 

Refine the partition ``self.part``. Set ``part_changed`` to ``True`` 

if and only if ``self.part`` was refined. 

  

OUTPUT: 

  

- ``False`` -- only if the actual node compares larger than the candidate 

for the canonical form. 

""" 

part_changed[0] = False 

cdef bint res, hyp_part_changed = not first_step 

cdef bint n_partition_changed = first_step 

cdef bint n_partition_changed_copy = n_partition_changed 

  

while hyp_part_changed or n_partition_changed: 

inner_group_changed = False 

res = self._inner_min_refine(&inner_group_changed, &n_partition_changed) 

if not res: 

return False 

  

part_changed[0] |= n_partition_changed 

n_partition_changed = n_partition_changed_copy 

n_partition_changed_copy = True 

  

if n_partition_changed: 

if PS_is_discrete(self._part): 

return True 

if inner_group_changed: 

continue 

  

while hyp_part_changed or n_partition_changed: 

if n_partition_changed: 

res = self._hyp_refine(&hyp_part_changed) 

if not res: 

return False 

n_partition_changed = False 

else: 

res = self._point_refine(&inner_group_changed, &n_partition_changed) 

if not res: 

return False 

part_changed[0] |= n_partition_changed 

hyp_part_changed = False 

if inner_group_changed: 

break # perform the inner_min_refine first! 

if n_partition_changed and PS_is_discrete(self._part): 

return True 

return True 

  

  

cdef bint _inner_min_refine(self, bint *inner_stab_changed, bint *changed_partition): 

""" 

Refine the partition ``self.part`` by computing the orbit (respectively 

the hash of a canonical form) of each column vector under the inner group. 

  

New fixed points of ``self.part`` get refined by the inner group. If this 

leads to a smaller group then we set ``inner_stab_changed`` to ``True``. 

``changed_partition`` is set to ``True`` if and only if ``self.part`` 

was refined. 

  

OUTPUT: 

  

- ``False`` only if the image under this homomorphism of group actions 

compares larger than the image of the candidate for the canonical form. 

""" 

cdef int i, j, res, stab_pow, apply_pow 

  

if self._inner_group.rank < 2: 

return True 

  

lower = iter(self._matrix[ : self._inner_group.rank ].columns()) 

upper = iter(self._matrix[ self._inner_group.rank : ].columns()) 

  

for i in range(self._n): 

l = next(lower) 

u = next(upper) 

  

if u.is_zero() and not i in self._fixed_minimized: 

# minimize by self._inner_group as in _inner_min: 

_, l = self._inner_group.minimize_by_row_mult(l) 

  

if self._inner_group.has_semilinear_action(): 

stab_pow = self._inner_group.frob_pow 

apply_pow = 0 

self._inner_group.minimize_by_frobenius(l, &apply_pow, &stab_pow) 

  

F = self._matrix.base_ring() 

f = F.hom([F.gen() ** (F.characteristic() ** apply_pow)]) 

l = l.apply_map(f) 

res = 0 

for r in iter(l): 

res *= self._q 

res += hash(r) 

self._refine_vals_scratch[i] = res 

else: 

self._refine_vals_scratch[i] = -1 

  

# provide some space to store the result (if not already exists) 

cdef long * best_vals = self._supp_refine_vals.get_row(self._nr_of_supp_refine_calls) 

self._nr_of_supp_refine_calls += 1 

return self._one_refinement(best_vals, 0, self._n, inner_stab_changed, 

changed_partition, "supp_refine") 

  

cdef bint _point_refine(self, bint *inner_stab_changed, bint *changed_partition): 

""" 

Refine the partition ``self.part`` by counting 

(colored) neighbours in the point-hyperplane graph. 

  

New fixed points of ``self.part`` get refined by the inner group. If this 

leads to a smaller group then we set ``inner_stab_changed`` to ``True``. 

``changed_partition`` is set to ``True`` if and only if ``self.part`` 

was refined. 

  

OUTPUT: 

  

- ``False`` only if the image under this homomorphism of group actions 

compares larger than the image of the candidate for the canonical form. 

""" 

  

self._part.depth += 1 

PS_clear(self._part) 

  

cdef bitset_t *nonsingletons = NULL 

cdef bitset_t scratch 

bitset_init(scratch, self._hyp_part.degree) 

cdef int nr_cells = PS_all_new_cells(self._hyp_part, & nonsingletons) 

  

for i in range(self._n): 

res = [0] * nr_cells 

for j in range(nr_cells): 

bitset_and(scratch, self._points2hyp[i], nonsingletons[j]) 

res[j] = bitset_hamming_weight(scratch) 

self._refine_vals_scratch[i] = hash(tuple(res)) 

  

for j in range(nr_cells): 

bitset_free(nonsingletons[j]) 

sig_free(nonsingletons) 

bitset_free(scratch) 

  

# provide some space to store the result (if not already exists) 

cdef long * best_vals = self._point_refine_vals.get_row(self._nr_of_point_refine_calls) 

self._nr_of_point_refine_calls += 1 

cdef bint ret_val = self._one_refinement(best_vals, 0, self._n, 

inner_stab_changed, changed_partition, "point_refine") 

  

if not changed_partition[0]: 

self._part.depth -= 1 

return ret_val 

  

cdef bint _hyp_refine(self, bint *changed_partition): 

""" 

Refine the partition of the hyperplanes by counting 

(colored) neighbours in the point-hyperplane graph. 

  

``changed_partition`` is set to ``True`` if and only if the partition 

was refined. 

  

OUTPUT: 

  

- ``False`` only if the image under this homomorphism of group actions 

compares larger than the image of the candidate for the canonical form. 

""" 

  

  

self._hyp_part.depth += 1 

PS_clear(self._hyp_part) 

cdef bitset_t *nonsingletons = NULL 

cdef bitset_t scratch 

bitset_init(scratch, self._part.degree) 

cdef int nr_cells = PS_all_new_cells(self._part, & nonsingletons) 

  

for i in range(self._hyp_part.degree): 

res = [0] * nr_cells 

for j in range(nr_cells): 

bitset_and(scratch, self._hyp2points[i], nonsingletons[j]) 

res[j] = bitset_hamming_weight(scratch) 

self._hyp_refine_vals_scratch[i] = hash(tuple(res)) 

  

for j in range(nr_cells): 

bitset_free(nonsingletons[j]) 

sig_free(nonsingletons) 

bitset_free(scratch) 

  

# provide some space to store the result (if not already exists) 

cdef long * best_vals = self._hyp_refine_vals.get_row(self._nr_of_hyp_refine_calls) 

self._nr_of_hyp_refine_calls += 1 

  

cdef tuple ret_val = PS_refinement(self._hyp_part, 

self._hyp_refine_vals_scratch, best_vals, 0, self._hyp_part.degree, 

&self._is_candidate_initialized, changed_partition) 

  

if not changed_partition[0]: 

self._hyp_part.depth -= 1 

return ret_val[0] 

  

cdef tuple _store_state_(self): 

r""" 

Store the current state of the node to a tuple, such that it can be 

restored by :meth:`_restore_state_`. 

""" 

return (self._inner_group._new_c(), self._nr_of_supp_refine_calls, 

self._nr_of_point_refine_calls, self._nr_of_hyp_refine_calls, 

self._hyp_part.depth) 

  

cdef void _restore_state_(self, tuple act_state): 

r""" 

The inverse of :meth:`_store_state_`. 

""" 

self._inner_group.copy_from(act_state[0]) 

self._nr_of_supp_refine_calls = act_state[1] 

self._nr_of_point_refine_calls = act_state[2] 

self._nr_of_hyp_refine_calls = act_state[3] 

self._hyp_part.depth = act_state[4] 

  

cdef void _store_best_(self): 

""" 

Store this node as the actual best candidate for the canonical form. 

""" 

self._best_candidate = copy(self._matrix) 

  

cdef void _latex_act_node(self, str comment="", int printlvl=0): 

""" 

Print the actual status as latex (tikz) commands to 

``self._latex_debug_string``. Only needed if one wants to visualize 

the algorithm using latex. This is very helpful for debugging purposes. 

""" 

if not BACKTRACK_WITHLATEX_DEBUG: 

return 

  

self._latex_debug_string += "\\node{\\begin{tabular}{" 

cdef int i, j, last = -1 

divide_sign = "" 

for i from 0 <= i < self._part.degree: 

if self._part.levels[i] <= self._part.depth: 

self._latex_debug_string += divide_sign + "*{" + str(i - last) + "}{c}" 

last = i 

divide_sign = "|" 

self._latex_debug_string += "}" 

  

# Print the applied permutation. We do highlight the fixed positions. 

for i from 0 <= i < (self._n): 

if self._part.entries[i] in self._fixed_minimized: 

self._latex_debug_string += "\\color{red}" 

  

self._latex_debug_string += str(self._part.entries[i]) 

if i == self._n - 1: 

self._latex_debug_string += " \\\\\\hline\n" 

else: 

self._latex_debug_string += " & " 

  

permuted_matrix = self._matrix.matrix_from_columns([self._part.entries[i] for i in range(self._n) ]) 

  

# Now we will finally print the matrix. 

for i from 0 <= i < self._k: 

for j from 0 <= j < (self._n - 1): 

self._latex_debug_string += "$" + permuted_matrix[i, j]._latex_() + "$ & " 

self._latex_debug_string += "$" + permuted_matrix[i, self._n - 1]._latex_() + "$ \\\\\n" 

  

if comment != "": 

self._latex_debug_string += "\\multicolumn{" + str(self._n) + "}{c}{" + comment + "}\n" 

self._latex_debug_string += "\\end{tabular}};\n"