Linear codes¶
Linear Codes¶
Let \(F = \GF{q}\) be a finite field. A rank \(k\) linear subspace of the vector space \(F^n\) is called an \([n, k]\)linear code, \(n\) being the length of the code and \(k\) its dimension. Elements of a code \(C\) are called codewords.
A linear map from \(F^k\) to an \([n,k]\) code \(C\) is called an “encoding”, and it can be represented as a \(k \times n\) matrix, called a generator matrix. Alternatively, \(C\) can be represented by its orthogonal complement in \(F^n\), i.e. the \((nk)\)dimensional vector space \(C^\perp\) such that the inner product of any element from \(C\) and any element from \(C^\perp\) is zero. \(C^\perp\) is called the dual code of \(C\), and any generator matrix for \(C^\perp\) is called a parity check matrix for \(C\).
We commonly endow \(F^n\) with the Hamming metric, i.e. the weight of a vector is the number of nonzero elements in it. The central operation of a linear code is then “decoding”: given a linear code \(C \subset F^n\) and a “received word” \(r \in F^n\) , retrieve the codeword \(c \in C\) such that the Hamming distance between \(r\) and \(c\) is minimal.
Families or Generic codes¶
Linear codes are either studied as generic vector spaces without any known structure, or as particular subfamilies with special properties.
The class sage.coding.linear_code.LinearCode
is used to represent the
former.
For the latter, these will be represented by specialised classes; for instance,
the family of Hamming codes are represented by the class
sage.coding.hamming_code.HammingCode
. Type codes.<tab>
for a list
of all code families known to Sage. Such code family classes should inherit from
the abstract base class sage.coding.linear_code.AbstractLinearCode
.
AbstractLinearCode
¶
This is a base class designed to contain methods, features and parameters shared by every linear code. For instance, generic algorithms for computing the minimum distance, the covering radius, etc. Many of these algorithms are slow, e.g. exponential in the code length. For specific subfamilies, better algorithms or even closed formulas might be known, in which case the respective method should be overridden.
AbstractLinearCode
is an abstract class for linear codes, so any linear code
class should inherit from this class. Also AbstractLinearCode
should never
itself be instantiated.
See sage.coding.linear_code.AbstractLinearCode
for details and
examples.
LinearCode
¶
This class is used to represent arbitrary and unstructured linear codes. It
mostly rely directly on generic methods provided by AbstractLinearCode
,
which means that basic operations on the code (e.g. computation of the minimum
distance) will use slow algorithms.
A LinearCode
is instantiated by providing a generator matrix:
sage: M = matrix(GF(2), [[1, 0, 0, 1, 0],\
[0, 1, 0, 1, 1],\
[0, 0, 1, 1, 1]])
sage: C = codes.LinearCode(M)
sage: C
[5, 3] linear code over GF(2)
sage: C.generator_matrix()
[1 0 0 1 0]
[0 1 0 1 1]
[0 0 1 1 1]
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.basis()
[
(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 = C.basis()[1]
sage: c in C
True
sage: c.nonzero_positions()
[0, 3, 4]
sage: c.support()
[0, 3, 4]
sage: c.parent()
Vector space of dimension 7 over Finite Field of size 2
Further references¶
If you want to get started on Sage’s linear codes library, see https://doc.sagemath.org/html/en/thematic_tutorials/coding_theory.html
If you want to learn more on the design of this library, see https://doc.sagemath.org/html/en/thematic_tutorials/structures_in_coding_theory.html
REFERENCES:
AUTHORS:
 David Joyner (20051122, 20061203): initial version
 William Stein (20060123): Inclusion in Sage
 David Joyner (20060130, 200604): small fixes
 David Joyner (200607): added documentation, grouptheoretical methods, ToricCode
 David Joyner (200608): hopeful latex fixes to documentation, added list and __iter__ methods to LinearCode and examples, added hamming_weight function, fixed random method to return a vector, TrivialCode, fixed subtle bug in dual_code, added galois_closure method, fixed mysterious bug in permutation_automorphism_group (GAP was overusing “G” somehow?)
 David Joyner (200608): hopeful latex fixes to documentation, added CyclicCode, best_known_linear_code, bounds_minimum_distance, assmus_mattson_designs (implementing AssmusMattson Theorem).
 David Joyner (200609): modified decode syntax, fixed bug in is_galois_closed, added LinearCode_from_vectorspace, extended_code, zeta_function
 Nick Alexander (20061210): factor GUAVA code to guava.py
 David Joyner (200705): added methods punctured, shortened, divisor, characteristic_polynomial, binomial_moment, support for LinearCode. Completely rewritten zeta_function (old version is now zeta_function2) and a new function, LinearCodeFromVectorSpace.
 David Joyner (200711): added zeta_polynomial, weight_enumerator, chinen_polynomial; improved best_known_code; made some pythonic revisions; added is_equivalent (for binary codes)
 David Joyner (200801): fixed bug in decode reported by Harald Schilly, (with Mike Hansen) added some doctests.
 David Joyner (200802): translated standard_form, dual_code to Python.
 David Joyner (200803): translated punctured, shortened, extended_code, random (and renamed random to random_element), deleted zeta_function2, zeta_function3, added wrapper automorphism_group_binary_code to Robert Miller’s code), added direct_sum_code, is_subcode, is_self_dual, is_self_orthogonal, redundancy_matrix, did some alphabetical reorganizing to make the file more readable. Fixed a bug in permutation_automorphism_group which caused it to crash.
 David Joyner (200803): fixed bugs in spectrum and zeta_polynomial, which misbehaved over nonprime base rings.
 David Joyner (200810): use CJ Tjhal’s MinimumWeight if char = 2 or 3 for min_dist; add is_permutation_equivalent and improve permutation_automorphism_group using an interface with Robert Miller’s code; added interface with Leon’s code for the spectrum method.
 David Joyner (200902): added native decoding methods (see module_decoder.py)
 David Joyner (200905): removed dependence on Guava, allowing it to be an option. Fixed errors in some docstrings.
 Kwankyu Lee (201001): added methods generator_matrix_systematic, information_set, and magma interface for linear codes.
 Niles Johnson (201008): trac ticket #3893:
random_element()
should pass on*args
and**kwds
.  Thomas Feulner (201211): trac ticket #13723: deprecation of
hamming_weight()
 Thomas Feulner (201310): added methods to compute a canonical representative and the automorphism group

class
sage.coding.linear_code.
AbstractLinearCode
(base_field, length, default_encoder_name, default_decoder_name)¶ Bases:
sage.coding.abstract_code.AbstractCode
,sage.modules.module.Module
Abstract base class for linear codes.
This class contains all methods that can be used on Linear Codes and on Linear Codes families. So, every Linear Coderelated 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 itslength
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 encoderMyEncoderClass
toMyNewCodeClass
under the nameMyEncoderName
. All I need to do is to write this line in the__init__.py
file:MyNewCodeClass._registered_encoders["NameOfMyEncoder"] = MyEncoderClass
and all instances ofMyNewCodeClass
will be able to use instances ofMyEncoderClass
.fill the dictionary of its decoders in
sage.coding.__init__
file. Example: I want to link the encoderMyDecoderClass
toMyNewCodeClass
under the nameMyDecoderName
. All I need to do is to write this line in the__init__.py
file:MyNewCodeClass._registered_decoders["NameOfMyDecoder"] = MyDecoderClass
and all instances ofMyNewCodeClass
will be able to use instances ofMyDecoderClass
.
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
AbstractLinearCode
has a generic implementation of the method__eq__
which uses the generator matrix and is quite slow. In subclasses you are encouraged to override__eq__
and__hash__
.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.

ambient_space
()¶ 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

assmus_mattson_designs
(t, mode=None)¶ 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 \([n, nk, d^*]\) code \(C^*\). Fix a \(t\), \(0<t<d\), and let
\[s = \{ i\ \ A_i^* \not= 0, 0< i \leq nt\}.\]Assume \(s\leq dt\).
 If \(A_i\not= 0\) and \(d\leq i\leq n\) then \(C_i = \{ c \in C\ \ wt(c) = i\}\) holds a simple tdesign.
 If \(A_i^*\not= 0\) and \(d*\leq i\leq nt\) then \(C_i^* = \{ c \in C^*\ \ wt(c) = i\}\) holds a simple tdesign.
A block design is a pair \((X,B)\), where \(X\) is a nonempty finite set of \(v>0\) elements called points, and \(B\) is a nonempty finite multiset of size b whose elements are called blocks, such that each block is a nonempty 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 5design. Similarly for its dual code \(C^*\) (of course \(C=C^*\) in this case, so this info is extraneous). The test fails to produce 6designs (ie, the hypotheses of the theorem fail to hold, not that the 6designs 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) 0 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

automorphism_group_gens
(equivalence='semilinear')¶ Return generators of the automorphism group of
self
.INPUT:
equivalence
(optional) – which defines the acting group, eitherpermutational
linear
semilinear
OUTPUT:
 generators of the automorphism group of
self
 the order of the automorphism group of
self
EXAMPLES:
Note, this result can depend on the PRNG state in libgap in a way that depends on which packages are loaded, so we must reseed GAP to ensure a consistent result for this example:
sage: libgap.set_seed(0) 0 sage: C = codes.HammingCode(GF(4, 'z'), 3) sage: C.automorphism_group_gens() ([((1, z, z + 1, z, z, 1, 1, z, z + 1, z, z, 1, z, z + 1, z, z, 1, z, z + 1, z, z); (1,5,18,7,11,8)(2,12,21)(3,20,14,10,19,15)(4,9)(13,17,16), Ring endomorphism of Finite Field in z of size 2^2 Defn: z > z + 1), ((1, 1, z, z + 1, z, z, z + 1, z + 1, z, 1, 1, z, z, z + 1, z + 1, 1, z, z, 1, z, z + 1); (2,11)(3,13)(4,14)(5,20)(6,17)(8,15)(16,19), 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)], 362880) sage: C.automorphism_group_gens(equivalence="linear") ([((z, 1, 1, z, z + 1, z, z, z + 1, z + 1, z + 1, 1, z + 1, z, z, 1, 1, 1, z, z, z + 1, z); (1,6)(2,20,9,16)(3,10,8,11)(4,15,21,5)(12,17)(13,14,19,18), Ring endomorphism of Finite Field in z of size 2^2 Defn: z > z), ((1, z, z + 1, z, z, z, z + 1, z + 1, 1, z, z, z, 1, z, 1, z + 1, z, z + 1, z, z + 1, 1); (1,15,20,5,8,6,12,14,13,7,16,11,19,3,21,4,9,10,18,17,2), 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); (1,19)(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); (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); (1,17)(2,8)(3,14)(4,10)(7,12)(9,19)(13,18)(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); (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)

base_field
()¶ 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

basis
()¶ Returns a basis of
self
.OUTPUT:
Sequence
 an immutable sequence whose universe is ambient space ofself
.
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

binomial_moment
(i)¶ Returns the ith binomial moment of the \([n,k,d]_q\)code \(C\):
\[B_i(C) = \sum_{S, S=i} \frac{q^{k_S}1}{q1}\]where \(k_S\) is the dimension of the shortened code \(C_{JS}\), \(J=[1,2,...,n]\). (The normalized binomial moment is \(b_i(C) = \binom(n,d+i)^{1}B_{d+i}(C)\).) In other words, \(C_{JS}\) 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) 0 sage: C.binomial_moment(4) # long time 35
Warning
This is slow.
REFERENCE:

canonical_representative
(equivalence='semilinear')¶ Compute a canonical orbit representative under the action of the semimonomial transformation group.
See
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, eitherpermutational
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

cardinality
()¶ Return the size of this code.
EXAMPLES:
sage: C = codes.HammingCode(GF(2), 3) sage: C.cardinality() 16 sage: len(C) 16

characteristic
()¶ Returns the characteristic of the base ring of
self
.EXAMPLES:
sage: C = codes.HammingCode(GF(2), 3) sage: C.characteristic() 2

characteristic_polynomial
()¶ 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

chinen_polynomial
()¶ 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.

construction_x
(other, aux)¶ Construction X applied to
self=C_1
,other=C_2
andaux=C_a
.other
must be a subcode ofself
.If \(C_1\) is a \([n, k_1, d_1]\) linear code and \(C_2\) is a \([n, k_2, d_2]\) linear code, then \(k_1 > k_2\) and \(d_1 < d_2\). \(C_a\) must be a \([n_a, k_a, d_a]\) linear code, such that \(k_a + k_2 = k_1\) and \(d_a + d_1 \leq d_2\).
The method will then return a \([n+n_a, k_1, d_a+d_1]\) linear code.
EXAMPLES:
sage: C = codes.BCHCode(GF(2),15,7) sage: C [15, 5] BCH Code over GF(2) with designed distance 7 sage: D = codes.BCHCode(GF(2),15,5) sage: D [15, 7] BCH Code over GF(2) with designed distance 5 sage: C.is_subcode(D) True sage: C.minimum_distance() 7 sage: D.minimum_distance() 5 sage: aux = codes.HammingCode(GF(2),2) sage: aux = aux.dual_code() sage: aux.minimum_distance() 2 sage: Cx = D.construction_x(C,aux) sage: Cx [18, 7] linear code over GF(2) sage: Cx.minimum_distance() 7

covering_radius
()¶ Return the minimal integer \(r\) such that any element in the ambient space of
self
has distance at most \(r\) to a codeword ofself
.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) 1 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

dimension
()¶ 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() 2

direct_sum
(other)¶ Direct sum of the codes
self
andother
Returns the code given by the direct sum of the codes
self
andother
, 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)

divisor
()¶ Returns the greatest common divisor of the weights of the nonzero codewords.
EXAMPLES:
sage: C = codes.GolayCode(GF(2)) sage: C.divisor() # Type II selfdual 4 sage: C = codes.QuadraticResidueCodeEvenPair(17,GF(2))[0] sage: C.divisor() 2

dual_code
()¶ Returns the dual code \(C^{\perp}\) of the code \(C\),
\[C^{\perp} = \{ v \in V\ \ v\cdot c = 0,\ \forall c \in C \}.\]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)

extended_code
()¶ Returns \(self\) as an extended code.
See documentation of
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)

galois_closure
(F0)¶ 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

generator_matrix
(encoder_name=None, **kwargs)¶ 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 ofself
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]

gens
()¶ 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)]

genus
()¶ Returns the “Duursma genus” of the code, \(\gamma_C = n+1kd\).
EXAMPLES:
sage: C1 = codes.HammingCode(GF(2), 3); C1 [7, 4] Hamming Code over GF(2) sage: C1.genus() 1 sage: C2 = codes.HammingCode(GF(4,"a"), 2); C2 [5, 3] Hamming Code over GF(4) sage: C2.genus() 0
Since all Hamming codes have minimum distance 3, these computations agree with the definition, \(n+1kd\).

information_set
()¶ 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)

is_galois_closed
()¶ Checks if
self
is equal to its Galois closure.EXAMPLES:
sage: C = codes.HammingCode(GF(4,"a"), 3) sage: C.is_galois_closed() False

is_information_set
(positions)¶ Return whether the given positions form an information set.
INPUT:
 A list of positions, i.e. integers in the range 0 to \(n1\) 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

is_permutation_automorphism
(g)¶ 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) 0 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) 1 sage: g = S8("(1,2,3,4)") sage: C.is_permutation_automorphism(g) 0

is_permutation_equivalent
(other, algorithm=None)¶ Returns
True
ifself
andother
are permutation equivalent codes andFalse
otherwise.The
algorithm="verbose"
option also returns a permutation (ifTrue
) sendingself
toother
.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

is_projective
()¶ 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() 8
A nonprojective code:
sage: C = codes.LinearCode(matrix(GF(2),[[1,0,1],[1,1,1]])) sage: C.is_projective() False

is_self_dual
()¶ Returns
True
if the code is selfdual (in the usual Hamming inner product) andFalse
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

is_self_orthogonal
()¶ Returns
True
if this code is selforthogonal andFalse
otherwise.A code is selforthogonal 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

is_subcode
(other)¶ Returns
True
ifself
is a subcode ofother
.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

juxtapose
(other)¶ Juxtaposition of
self
andother
The two codes must have equal dimension.
EXAMPLES:
sage: C1 = codes.HammingCode(GF(2), 3) sage: C2 = C1.juxtapose(C1) sage: C2 [14, 4] linear code over GF(2)

length
()¶ Returns the length of this code.
EXAMPLES:
sage: C = codes.HammingCode(GF(2), 3) sage: C.length() 7

minimum_distance
(algorithm=None)¶ 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 toNone
, 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() 3
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") 3 sage: C.minimum_distance(algorithm="guava") # optional  gap_packages (Guava package) 3

module_composition_factors
(gp)¶ 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.

parity_check_matrix
()¶ 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]

permutation_automorphism_group
(algorithm='partition')¶ 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
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 codetheoretic 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 ofself
(callcanonical_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")
.

permuted_code
(p)¶ Returns the permuted code, which is equivalent to
self
via the column permutationp
.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

product_code
(other)¶ Combines
self
withother
to give the tensor product code.If
self
is a \([n_1, k_1, d_1]\)code andother
is a \([n_2, k_2, d_2]\)code, the product is a \([n_1n_2, k_1k_2, d_1d_2]\)code.Note that the two codes have to be over the same field.
EXAMPLES:
sage: C = codes.HammingCode(GF(2), 3) sage: C [7, 4] Hamming Code over GF(2) sage: D = codes.ReedMullerCode(GF(2), 2, 2) sage: D Binary ReedMuller Code of order 2 and number of variables 2 sage: A = C.product_code(D) sage: A [28, 16] linear code over GF(2) sage: A.length() == C.length()*D.length() True sage: A.dimension() == C.dimension()*D.dimension() True sage: A.minimum_distance() == C.minimum_distance()*D.minimum_distance() True

punctured
(L)¶ Returns a
sage.coding.punctured_code
object fromL
.INPUT:
L
 List of positions to puncture
OUTPUT:
 an instance of
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]

rate
()¶ 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

redundancy_matrix
()¶ Returns the nonidentity 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
AbstractLinearCode.systematic_generator_matrix()
.OUTPUT:
 An \(k \times (nk)\) 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]

relative_distance
()¶ 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

shortened
(L)¶ 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)

spectrum
(algorithm=None)¶ 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 nonnegative 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

standard_form
(return_permutation=True)¶ Returns a linear code which is permutationequivalent 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
) ifTrue
, the column permutation which bringsself
into the returned code is also returned.
OUTPUT:
 A
LinearCode
whosesystematic_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]

support
()¶ 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]

syndrome
(r)¶ Returns the syndrome of
r
.The syndrome of
r
is the result of \(H \times r\) where \(H\) is the parity check matrix ofself
. Ifr
belongs toself
, its syndrome equals to the zero vector.INPUT:
r
– a vector of the same length asself
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)

systematic_generator_matrix
(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 forLinearCodeSystematicEncoder
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]

u_u_plus_v_code
(other)¶ The \((uu+v)\)construction with
self=u
andother=v
Returns the code obtained through \((uu+v)\)construction with
self
as \(u\) andother
as \(v\). Note that \(u\) and \(v\) must have equal lengths. For \(u\) a \([n, k_1, d_1]\)code and \(v\) a \([n, k_2, d_2]\)code this returns a \([2n, k_1+k_2, d]\)code, where \(d=\min(2d_1,d_2)\).EXAMPLES:
sage: C1 = codes.HammingCode(GF(2), 3) sage: C2 = codes.HammingCode(GF(2), 3) sage: D = C1.u_u_plus_v_code(C2) sage: D [14, 8] linear code over GF(2)

weight_distribution
(algorithm=None)¶ 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 nonnegative 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

weight_enumerator
(names=None, bivariate=True)¶ Return the weight enumerator polynomial of
self
.This is the bivariate, homogeneous polynomial in \(x\) and \(y\) whose coefficient to \(x^i y^{ni}\) 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.bivariate
 (default: \(True\)) Whether to return a bivariate, homogeneous polynomial or just a univariate polynomial. If set toFalse
, thennames
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 nonsymmetrical 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

zero
()¶ 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)

zeta_function
(name='T')¶ 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() (1/5*T^2 + 1/5*T + 1/10)/(T^2  3/2*T + 1/2)

zeta_polynomial
(name='T')¶ 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) 3 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() 1
REFERENCES:

class
sage.coding.linear_code.
LinearCode
(generator, d=None)¶ Bases:
sage.coding.linear_code.AbstractLinearCode
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 familyspecific methods.
See Wikipedia article 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 asrank
); or a code over a finite fieldd
– (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() 4 sage: C.length() 7 sage: C.minimum_distance() 3 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() 3
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 ticket #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 (112005)
 Charles Prior (032016): trac ticket #20198, LinearCode from a code

generator_matrix
(encoder_name=None, **kwargs)¶ 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]

class
sage.coding.linear_code.
LinearCodeGeneratorMatrixEncoder
(code)¶ Bases:
sage.coding.encoder.Encoder
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
LinearCode
.INPUT:
code
– The associatedLinearCode
of this encoder.

generator_matrix
()¶ 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]

class
sage.coding.linear_code.
LinearCodeNearestNeighborDecoder
(code)¶ Bases:
sage.coding.decoder.Decoder
Construct a decoder for Linear Codes. This decoder will decode to the nearest codeword found.
INPUT:
code
– A code associated to this decoder

decode_to_code
(r)¶ Corrects the errors in
word
and returns a codeword.INPUT:
r
– a codeword ofself
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)

decoding_radius
()¶ 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() 1

class
sage.coding.linear_code.
LinearCodeSyndromeDecoder
(code, maximum_error_weight=None)¶ Bases:
sage.coding.decoder.Decoder
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 ofr
is computed. The corresponding error pattern is recovered from the precomputed 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 lowerweight errors with the same syndrome. If one setsmaximum_error_weight
to a value greater than the covering radius, then the covering radius will be determined while building the lookuptable. This lower value is then returned if you querydecoding_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 decodermaximum_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 \(nk\), since this is an upper bound on the covering radius on any linear code. Ifmaximum_error_weight
is kept unspecified, it will be set to \(n  k\), where \(n\) is the length ofcode
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 ofcode
. 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() {'alwayssucceed', 'bounded_distance', 'harddecision'} sage: D.decoding_radius() 1
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', 'harddecision', 'mighterror'} sage: D.decoding_radius() 3
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', 'harddecision', 'mighterror'} sage: D.decoding_radius() 4
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.
decode_to_code
(r)¶ Corrects the errors in
word
and returns a codeword.INPUT:
r
– a codeword ofself
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

decoding_radius
()¶ 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() 4

maximum_error_weight
()¶ 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() 4

syndrome_table
()¶ 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)}
 First, a lookup table is built by computing the syndrome of every error
pattern of weight up to

class
sage.coding.linear_code.
LinearCodeSystematicEncoder
(code, systematic_positions=None)¶ Bases:
sage.coding.encoder.Encoder
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
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
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 \(n1\) where \(n\) is the length of the code and \(k\) its dimension. The 0th symbol of a message will then be at positionsystematic_positions[0]
, the 1st index at positionsystematic_positions[1]
, etc. AValueError
is raised at construction time if the supplied indices do not form an information set.
EXAMPLES:
The following demonstrates the basic usage of
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
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
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

generator_matrix
()¶ 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
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]

systematic_permutation
()¶ Returns a permutation which would take the systematic positions into [0,..,k1]
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]

systematic_positions
()¶ 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)