# 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 $$(n-k)$$-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 non-zero 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 sub-families 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 (2005-11-22, 2006-12-03): initial version
• William Stein (2006-01-23): Inclusion in Sage
• David Joyner (2006-01-30, 2006-04): small fixes
• David Joyner (2006-07): added documentation, group-theoretical methods, ToricCode
• David Joyner (2006-08): 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 over-using “G” somehow?)
• David Joyner (2006-08): hopeful latex fixes to documentation, added CyclicCode, best_known_linear_code, bounds_minimum_distance, assmus_mattson_designs (implementing Assmus-Mattson Theorem).
• David Joyner (2006-09): modified decode syntax, fixed bug in is_galois_closed, added LinearCode_from_vectorspace, extended_code, zeta_function
• Nick Alexander (2006-12-10): factor GUAVA code to guava.py
• David Joyner (2007-05): 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 (2007-11): added zeta_polynomial, weight_enumerator, chinen_polynomial; improved best_known_code; made some pythonic revisions; added is_equivalent (for binary codes)
• David Joyner (2008-01): fixed bug in decode reported by Harald Schilly, (with Mike Hansen) added some doctests.
• David Joyner (2008-02): translated standard_form, dual_code to Python.
• David Joyner (2008-03): 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 (2008-03): fixed bugs in spectrum and zeta_polynomial, which misbehaved over non-prime base rings.
• David Joyner (2008-10): 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 (2009-02): added native decoding methods (see module_decoder.py)
• David Joyner (2009-05): removed dependence on Guava, allowing it to be an option. Fixed errors in some docstrings.
• Kwankyu Lee (2010-01): added methods generator_matrix_systematic, information_set, and magma interface for linear codes.
• Niles Johnson (2010-08): trac ticket #3893: random_element() should pass on *args and **kwds.
• Thomas Feulner (2012-11): trac ticket #13723: deprecation of hamming_weight()
• Thomas Feulner (2013-10): 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)

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 Code-related class should inherit from this abstract class.

To implement a linear code, you need to:

• inherit from AbstractLinearCode

• call AbstractLinearCode __init__ method in the subclass constructor. Example: super(SubclassName, self).__init__(base_field, length, "EncoderName", "DecoderName"). By doing that, your subclass will have its length parameter initialized and will be properly set as a member of the category framework. You need of course to complete the constructor by adding any additional parameter needed to describe properly the code defined in the subclass.

• Add the following two lines on the class level:

_registered_encoders = {}
_registered_decoders = {}

• fill the dictionary of its encoders in sage.coding.__init__.py file. Example: I want to link the encoder MyEncoderClass to MyNewCodeClass under the name MyEncoderName. All I need to do is to write this line in the __init__.py file: MyNewCodeClass._registered_encoders["NameOfMyEncoder"] = MyEncoderClass and all instances of MyNewCodeClass will be able to use instances of MyEncoderClass.

• fill the dictionary of its decoders in sage.coding.__init__ file. Example: I want to link the encoder MyDecoderClass to MyNewCodeClass under the name MyDecoderName. All I need to do is to write this line in the __init__.py file: MyNewCodeClass._registered_decoders["NameOfMyDecoder"] = MyDecoderClass and all instances of MyNewCodeClass will be able to use instances of MyDecoderClass.

As AbstractLinearCode is not designed to be implemented, it does not have any representation methods. You should implement _repr_ and _latex_ methods in the subclass.

Note

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.

add_decoder(name, decoder)

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

Note

This method only adds decoder to self, and not to any member of the class of self. To know how to add an sage.coding.decoder.Decoder, please refer to the documentation of AbstractLinearCode.

INPUT:

• name – the string name for the decoder
• decoder – the class name of the decoder

EXAMPLES:

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

sage: class MyDecoder(sage.coding.decoder.Decoder):
....:   def __init__(self, code):
....:       super(MyDecoder, self).__init__(code)
....:   def _repr_(self):
....:       return "MyDecoder decoder with associated code %s" % self.code()


We now create a new code:

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


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

sage: C.add_decoder("MyDecoder", MyDecoder)
sage: sorted(C.decoders_available())
['InformationSet', 'MyDecoder', 'NearestNeighbor', 'Syndrome']


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

sage: C2 = codes.HammingCode(GF(2), 3)
sage: sorted(C2.decoders_available())
['InformationSet', 'NearestNeighbor', 'Syndrome']

add_encoder(name, encoder)

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

Note

This method only adds encoder to self, and not to any member of the class of self. To know how to add an sage.coding.encoder.Encoder, please refer to the documentation of AbstractLinearCode.

INPUT:

• name – the string name for the encoder
• encoder – the class name of the encoder

EXAMPLES:

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

sage: class MyEncoder(sage.coding.encoder.Encoder):
....:   def __init__(self, code):
....:       super(MyEncoder, self).__init__(code)
....:   def _repr_(self):
....:       return "MyEncoder encoder with associated code %s" % self.code()


We now create a new code:

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


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

sage: C.add_encoder("MyEncoder", MyEncoder)
sage: sorted(C.encoders_available())
['MyEncoder', 'Systematic']


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

sage: C2 = codes.HammingCode(GF(2), 3)
sage: sorted(C2.encoders_available())
['Systematic']

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 $$[n, n-k, d^*]$$ code $$C^*$$. Fix a $$t$$, $$0<t<d$$, and let

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

Assume $$s\leq d-t$$.

1. If $$A_i\not= 0$$ and $$d\leq i\leq n$$ then $$C_i = \{ c \in C\ |\ wt(c) = i\}$$ holds a simple t-design.
2. If $$A_i^*\not= 0$$ and $$d*\leq i\leq n-t$$ then $$C_i^* = \{ c \in C^*\ |\ wt(c) = i\}$$ holds a simple t-design.

A block design is a pair $$(X,B)$$, where $$X$$ is a non-empty finite set of $$v>0$$ elements called points, and $$B$$ is a non-empty finite multiset of size b whose elements are called blocks, such that each block is a non-empty finite multiset of $$k$$ points. $$A$$ design without repeated blocks is called a simple block design. If every subset of points of size $$t$$ is contained in exactly $$\lambda$$ blocks the block design is called a $$t-(v,k,\lambda)$$ design (or simply a $$t$$-design when the parameters are not specified). When $$\lambda=1$$ then the block design is called a $$S(t,k,v)$$ Steiner system.

In the Assmus and Mattson Theorem (1), $$X$$ is the set $$\{1,2,...,n\}$$ of coordinate locations and $$B = \{supp(c)\ |\ c \in C_i\}$$ is the set of supports of the codewords of $$C$$ of weight $$i$$. Therefore, the parameters of the $$t$$-design for $$C_i$$ are

t =       given
v =       n
k =       i   (k not to be confused with dim(C))
b =       Ai
lambda = b*binomial(k,t)/binomial(v,t) (by Theorem 8.1.6,
p 294, in [HP2003]_)


Setting the mode="verbose" option prints out the values of the parameters.

The first example below means that the binary [24,12,8]-code C has the property that the (support of the) codewords of weight 8 (resp., 12, 16) form a 5-design. Similarly for its dual code $$C^*$$ (of course $$C=C^*$$ in this case, so this info is extraneous). The test fails to produce 6-designs (ie, the hypotheses of the theorem fail to hold, not that the 6-designs definitely don’t exist). The command assmus_mattson_designs(C,5,mode=”verbose”) returns the same value but prints out more detailed information.

The second example below illustrates the blocks of the 5-(24, 8, 1) design (i.e., the S(5,8,24) Steiner system).

EXAMPLES:

sage: C = codes.GolayCode(GF(2))             #  example 1
sage: C.assmus_mattson_designs(5)
['weights from C: ',
[8, 12, 16, 24],
'designs from C: ',
[[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)], [5, (24, 24, 1)]],
'weights from C*: ',
[8, 12, 16],
'designs from C*: ',
[[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)]]]
sage: C.assmus_mattson_designs(6)
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, either

• permutational
• 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 re-seed 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 of self.

EXAMPLES:

sage: C = codes.HammingCode(GF(2), 3)
sage: C.basis()
[
(1, 0, 0, 0, 0, 1, 1),
(0, 1, 0, 0, 1, 0, 1),
(0, 0, 1, 0, 1, 1, 0),
(0, 0, 0, 1, 1, 1, 1)
]
sage: C.basis().universe()
Vector space of dimension 7 over Finite Field of size 2

binomial_moment(i)

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

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

where $$k_S$$ is the dimension of the shortened code $$C_{J-S}$$, $$J=[1,2,...,n]$$. (The normalized binomial moment is $$b_i(C) = \binom(n,d+i)^{-1}B_{d+i}(C)$$.) In other words, $$C_{J-S}$$ is isomorphic to the subcode of C of codewords supported on S.

EXAMPLES:

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

• permutational
• linear
• semilinear

OUTPUT:

• a canonical representative of self
• a semimonomial transformation mapping self onto its representative

EXAMPLES:

sage: F.<z> = GF(4)
sage: C = codes.HammingCode(F, 3)
sage: CanRep, transp = C.canonical_representative()


Check that the transporter element is correct:

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


Check if an equivalent code has the same canonical representative:

sage: f = F.hom([z**2])
sage: C_iso = LinearCode(C.generator_matrix().apply_map(f))
sage: CanRep_iso, _ = C_iso.canonical_representative()
sage: CanRep_iso == CanRep
True


Since applying the Frobenius automorphism could be extended to an automorphism of $$C$$, the following must also yield True:

sage: CanRep1, _ = C.canonical_representative("linear")
sage: CanRep2, _ = C_iso.canonical_representative("linear")
sage: CanRep2 == CanRep1
True

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 and aux=C_a.

other must be a subcode of self.

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 of self.

This method requires the optional GAP package Guava.

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

Note

This method is currently not implemented on codes over base fields of cardinality greater than 256 due to limitations in the underlying algorithm of GAP.

EXAMPLES:

sage: C = codes.HammingCode(GF(2), 5)
sage: C.covering_radius()  # optional - gap_packages (Guava package)
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

decode_to_code(word, decoder_name=None, *args, **kwargs)

Corrects the errors in word and returns a codeword.

INPUT:

• word – a vector of the same length as self over the base field of self
• decoder_name – (default: None) Name of the decoder which will be used to decode word. The default decoder of self will be used if default value is kept.
• args, kwargs – all additional arguments are forwarded to decoder()

OUTPUT:

• A vector of self.

EXAMPLES:

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])
sage: C = LinearCode(G)
sage: word = vector(GF(2), (1, 1, 0, 0, 1, 1, 0))
sage: w_err = word + vector(GF(2), (1, 0, 0, 0, 0, 0, 0))
sage: C.decode_to_code(w_err)
(1, 1, 0, 0, 1, 1, 0)


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

sage: sorted(C.decoders_available())
['InformationSet', 'NearestNeighbor', 'Syndrome']
sage: C.decode_to_code(w_err, 'NearestNeighbor')
(1, 1, 0, 0, 1, 1, 0)

decode_to_message(word, decoder_name=None, *args, **kwargs)

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

INPUT:

• word – a vector of the same length as self over the base field of self
• decoder_name – (default: None) Name of the decoder which will be used to decode word. The default decoder of self will be used if default value is kept.
• args, kwargs – all additional arguments are forwarded to decoder()

OUTPUT:

• A vector of the message space of self.

EXAMPLES:

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])
sage: C = LinearCode(G)
sage: word = vector(GF(2), (1, 1, 0, 0, 1, 1, 0))
sage: C.decode_to_message(word)
(0, 1, 1, 0)


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

sage: sorted(C.decoders_available())
['InformationSet', 'NearestNeighbor', 'Syndrome']
sage: C.decode_to_message(word, 'NearestNeighbor')
(0, 1, 1, 0)

decoder(decoder_name=None, *args, **kwargs)

Return a decoder of self.

INPUT:

• decoder_name – (default: None) name of the decoder which will be returned. The default decoder of self will be used if default value is kept.
• args, kwargs – all additional arguments will be forwarded to the constructor of the decoder that will be returned by this method

OUTPUT:

• a decoder object

Besides creating the decoder and returning it, this method also stores the decoder in a cache. With this behaviour, each decoder will be created at most one time for self.

EXAMPLES:

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])
sage: C = LinearCode(G)
sage: C.decoder()
Syndrome decoder for [7, 4] linear code over GF(2) handling errors of weight up to 1


If the name of a decoder which is not known by self is passed, an exception will be raised:

sage: sorted(C.decoders_available())
['InformationSet', 'NearestNeighbor', 'Syndrome']
sage: C.decoder('Try')
Traceback (most recent call last):
...
ValueError: There is no Decoder named 'Try'. The known Decoders are: ['InformationSet', 'NearestNeighbor', 'Syndrome']


Some decoders take extra arguments. If the user forgets to supply these, the error message attempts to be helpful:

sage: C.decoder('InformationSet')
Traceback (most recent call last):
...
ValueError: Constructing the InformationSet decoder failed, possibly due to missing or incorrect parameters.
The constructor requires the arguments ['number_errors'].
It takes the optional arguments ['algorithm'].
It accepts unspecified arguments as well.
See the documentation of sage.coding.information_set_decoder.LinearCodeInformationSetDecoder for more details.

decoders_available(classes=False)

Returns a list of the available decoders’ names for self.

INPUT:

• classes – (default: False) if classes is set to True, return instead a dict mapping available decoder name to the associated decoder class.

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

EXAMPLES:

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])
sage: C = LinearCode(G)
sage: C.decoders_available()
['InformationSet', 'NearestNeighbor', 'Syndrome']

sage: dictionary = C.decoders_available(True)
sage: sorted(dictionary.keys())
['InformationSet', 'NearestNeighbor', 'Syndrome']
sage: dictionary['NearestNeighbor']
<class 'sage.coding.linear_code.LinearCodeNearestNeighborDecoder'>

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 and other

Returns the code given by the direct sum of the codes self and other, which must be linear codes defined over the same base ring.

EXAMPLES:

sage: C1 = codes.HammingCode(GF(2), 3)
sage: C2 = C1.direct_sum(C1); C2
[14, 8] linear code over GF(2)
sage: C3 = C1.direct_sum(C2); C3
[21, 12] linear code over GF(2)

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 self-dual
4
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)

encode(word, encoder_name=None, *args, **kwargs)

Transforms an element of a message space into a codeword.

INPUT:

• word – a vector of a message space of the code.
• encoder_name – (default: None) Name of the encoder which will be used to encode word. The default encoder of self will be used if default value is kept.
• args, kwargs – all additional arguments are forwarded to the construction of the encoder that is used.

Note

The default encoder always has $$F^{k}$$ as message space, with $$k$$ the dimension of self and $$F$$ the base ring of self.

One can use the following shortcut to encode a word

C(word)


OUTPUT:

• a vector of self.

EXAMPLES:

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])
sage: C = LinearCode(G)
sage: word = vector((0, 1, 1, 0))
sage: C.encode(word)
(1, 1, 0, 0, 1, 1, 0)
sage: C(word)
(1, 1, 0, 0, 1, 1, 0)


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

sage: sorted(C.encoders_available())
['GeneratorMatrix', 'Systematic']
sage: word = vector((0, 1, 1, 0))
sage: C.encode(word, 'GeneratorMatrix')
(1, 1, 0, 0, 1, 1, 0)

encoder(encoder_name=None, *args, **kwargs)

Returns an encoder of self.

The returned encoder provided by this method is cached.

This methods creates a new instance of the encoder subclass designated by encoder_name. While it is also possible to do the same by directly calling the subclass’ constructor, it is strongly advised to use this method to take advantage of the caching mechanism.

INPUT:

• encoder_name – (default: None) name of the encoder which will be returned. The default encoder of self will be used if default value is kept.
• args, kwargs – all additional arguments are forwarded to the constructor of the encoder this method will return.

OUTPUT:

• an Encoder object.

Note

The default encoder always has $$F^{k}$$ as message space, with $$k$$ the dimension of self and $$F$$ the base ring of self.

EXAMPLES:

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])
sage: C = LinearCode(G)
sage: C.encoder()
Generator matrix-based encoder for [7, 4] linear code over GF(2)


We check that the returned encoder is cached:

sage: C.encoder.is_in_cache()
True


If the name of an encoder which is not known by self is passed, an exception will be raised:

sage: sorted(C.encoders_available())
['GeneratorMatrix', 'Systematic']
sage: C.encoder('NonExistingEncoder')
Traceback (most recent call last):
...
ValueError: There is no Encoder named 'NonExistingEncoder'. The known Encoders are: ['GeneratorMatrix', 'Systematic']


Some encoders take extra arguments. If the user incorrectly supplies these, the error message attempts to be helpful:

sage: C.encoder('Systematic', strange_parameter=True)
Traceback (most recent call last):
...
ValueError: Constructing the Systematic encoder failed, possibly due to missing or incorrect parameters.
The constructor requires no arguments.
It takes the optional arguments ['systematic_positions'].
See the documentation of sage.coding.linear_code.LinearCodeSystematicEncoder for more details.

encoders_available(classes=False)

Returns a list of the available encoders’ names for self.

INPUT:

• classes – (default: False) if classes is set to True, return instead a dict mapping available encoder name to the associated encoder class.

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

EXAMPLES:

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])
sage: C = LinearCode(G)
sage: C.encoders_available()
['GeneratorMatrix', 'Systematic']
sage: dictionary = C.encoders_available(True)
sage: sorted(dictionary.items())
[('GeneratorMatrix', <class 'sage.coding.linear_code.LinearCodeGeneratorMatrixEncoder'>),
('Systematic', <class 'sage.coding.linear_code.LinearCodeSystematicEncoder'>)]

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 of self will be used if default value is kept.
• kwargs – all additional arguments are forwarded to the construction of the encoder that is used.

EXAMPLES:

sage: G = matrix(GF(3),2,[1,-1,1,-1,1,1])
sage: code = LinearCode(G)
sage: code.generator_matrix()
[1 2 1]
[2 1 1]

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, $$\gamma_C = n+1-k-d$$.

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, $$n+1-k-d$$.

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 $$n-1$$ where $$n$$ is the length of $$self$$.

OUTPUT:

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

EXAMPLES:

sage: G = matrix(GF(3),2,[1,2,0,                                      2,1,1])
sage: code = LinearCode(G)
sage: code.is_information_set([0,1])
False
sage: code.is_information_set([0,2])
True

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 if self and other are permutation equivalent codes and False otherwise.

The algorithm="verbose" option also returns a permutation (if True) sending self to other.

Uses Robert Miller’s double coset partition refinement work.

EXAMPLES:

sage: P.<x> = PolynomialRing(GF(2),"x")
sage: g = x^3+x+1
sage: C1 = codes.CyclicCode(length = 7, generator_pol = g); C1
[7, 4] Cyclic Code over GF(2)
sage: C2 = codes.HammingCode(GF(2), 3); C2
[7, 4] Hamming Code over GF(2)
sage: C1.is_permutation_equivalent(C2)
True
sage: C1.is_permutation_equivalent(C2,algorithm="verbose")
(True, (3,4)(5,7,6))
sage: C1 = codes.random_linear_code(GF(2), 10, 5)
sage: C2 = codes.random_linear_code(GF(3), 10, 5)
sage: C1.is_permutation_equivalent(C2)
False

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 non-projective 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 self-dual (in the usual Hamming inner product) and False otherwise.

EXAMPLES:

sage: C = codes.GolayCode(GF(2))
sage: C.is_self_dual()
True
sage: C = codes.HammingCode(GF(2), 3)
sage: C.is_self_dual()
False

is_self_orthogonal()

Returns True if this code is self-orthogonal and False otherwise.

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

EXAMPLES:

sage: C = codes.GolayCode(GF(2))
sage: C.is_self_orthogonal()
True
sage: C = codes.HammingCode(GF(2), 3)
sage: C.is_self_orthogonal()
False
sage: C = codes.QuasiQuadraticResidueCode(11)  # optional - gap_packages (Guava package)
sage: C.is_self_orthogonal()             # optional - gap_packages (Guava package)
True

is_subcode(other)

Returns True if self is a subcode of other.

EXAMPLES:

sage: C1 = codes.HammingCode(GF(2), 3)
sage: G1 = C1.generator_matrix()
sage: G2 = G1.matrix_from_rows([0,1,2])
sage: C2 = LinearCode(G2)
sage: C2.is_subcode(C1)
True
sage: C1.is_subcode(C2)
False
sage: C3 = C1.extended_code()
sage: C1.is_subcode(C3)
False
sage: C4 = C1.punctured([1])
sage: C4.is_subcode(C1)
False
sage: C5 = C1.shortened([1])
sage: C5.is_subcode(C1)
False
sage: C1 = codes.HammingCode(GF(9,"z"), 3)
sage: G1 = C1.generator_matrix()
sage: G2 = G1.matrix_from_rows([0,1,2])
sage: C2 = LinearCode(G2)
sage: C2.is_subcode(C1)
True

juxtapose(other)

Juxtaposition of self and other

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

list()

Return a list of all elements of this linear code.

EXAMPLES:

sage: C = codes.HammingCode(GF(2), 3)
sage: Clist = C.list()
sage: Clist[5]; Clist[5] in C
(1, 0, 1, 0, 1, 0, 1)
True

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 to None, GAP methods will be used. algorithm can be: - "Guava", which will use optional GAP package Guava

OUTPUT:

• Integer, minimum distance of this code

EXAMPLES:

sage: MS = MatrixSpace(GF(3),4,7)
sage: G = MS([[1,1,1,0,0,0,0], [1,0,0,1,1,0,0], [0,1,0,1,0,1,0], [1,1,0,1,0,0,1]])
sage: C = LinearCode(G)
sage: C.minimum_distance()
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 code-theoretic data is printed out at several stages of the computation. If "partition" then the (default) partition refinement algorithm of Robert Miller is used. Finally, if "codecan" then the partition refinement algorithm of Thomas Feulner is used, which also computes a canonical representative of self (call canonical_representative() to access it).

OUTPUT:

• Permutation automorphism group

EXAMPLES:

sage: MS = MatrixSpace(GF(2),4,8)
sage: G  = MS([[1,0,0,0,1,1,1,0],[0,1,1,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0]])
sage: C  = LinearCode(G)
sage: C
[8, 4] linear code over GF(2)
sage: G = C.permutation_automorphism_group()
sage: G.order()
144
sage: GG = C.permutation_automorphism_group("codecan")
sage: GG == G
True


A less easy example involves showing that the permutation automorphism group of the extended ternary Golay code is the Mathieu group $$M_{11}$$.

sage: C = codes.GolayCode(GF(3))
sage: M11 = MathieuGroup(11)
sage: M11.order()
7920
sage: G = C.permutation_automorphism_group()  # long time (6s on sage.math, 2011)
sage: G.is_isomorphic(M11)                    # long time
True
sage: GG = C.permutation_automorphism_group("codecan") # long time
sage: GG == G # long time
True


Other examples:

sage: C = codes.GolayCode(GF(2))
sage: G = C.permutation_automorphism_group()
sage: G.order()
244823040
sage: C = codes.HammingCode(GF(2), 5)
sage: G = C.permutation_automorphism_group()
sage: G.order()
9999360
sage: C = codes.HammingCode(GF(3), 2); C
[4, 2] Hamming Code over GF(3)
sage: C.permutation_automorphism_group(algorithm="partition")
Permutation Group with generators [(1,3,4)]
sage: C = codes.HammingCode(GF(4,"z"), 2); C
[5, 3] Hamming Code over GF(4)
sage: G = C.permutation_automorphism_group(algorithm="partition"); G
Permutation Group with generators [(1,3)(4,5), (1,4)(3,5)]
sage: GG = C.permutation_automorphism_group(algorithm="codecan") # long time
sage: GG == G # long time
True
sage: C.permutation_automorphism_group(algorithm="gap")  # optional - gap_packages (Guava package)
Permutation Group with generators [(1,3)(4,5), (1,4)(3,5)]
sage: C = codes.GolayCode(GF(3), True)
sage: C.permutation_automorphism_group(algorithm="gap")  # optional - gap_packages (Guava package)
Permutation Group with generators [(5,7)(6,11)(8,9)(10,12), (4,6,11)(5,8,12)(7,10,9), (3,4)(6,8)(9,11)(10,12), (2,3)(6,11)(8,12)(9,10), (1,2)(5,10)(7,12)(8,9)]


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

Minimum distance: 5 Weight distribution: [1, 0, 0, 0, 0, 132, 132,
0, 330, 110, 0, 24]

Using the 132 codewords of weight 5 Supergroup size: 39916800


in addition to the output of C.permutation_automorphism_group(algorithm="gap").

permuted_code(p)

Returns the permuted code, which is equivalent to self via the column permutation p.

EXAMPLES:

sage: C = codes.HammingCode(GF(2), 3)
sage: G = C.permutation_automorphism_group(); G
Permutation Group with generators [(4,5)(6,7), (4,6)(5,7), (2,3)(6,7), (2,4)(3,5), (1,2)(5,6)]
sage: g = G("(2,3)(6,7)")
sage: Cg = C.permuted_code(g)
sage: Cg
[7, 4] linear code over GF(2)
sage: C.generator_matrix() == Cg.systematic_generator_matrix()
True

product_code(other)

Combines self with other to give the tensor product code.

If self is a $$[n_1, k_1, d_1]$$-code and other 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 Reed-Muller 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 from L.

INPUT:

• L - List of positions to puncture

OUTPUT:

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]

random_element(*args, **kwds)

Returns a random codeword; passes other positional and keyword arguments to random_element() method of vector space.

OUTPUT:

• Random element of the vector space of this code

EXAMPLES:

sage: C = codes.HammingCode(GF(4,'a'), 3)
sage: C.random_element() # random test
(1, 0, 0, a + 1, 1, a, a, a + 1, a + 1, 1, 1, 0, a + 1, a, 0, a, a, 0, a, a, 1)


Passes extra positional or keyword arguments through:

sage: C.random_element(prob=.5, distribution='1/n') # random test
(1, 0, a, 0, 0, 0, 0, a + 1, 0, 0, 0, 0, 0, 0, 0, 0, a + 1, a + 1, 1, 0, 0)

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 non-identity columns of a systematic generator matrix for self.

A systematic generator matrix is a generator matrix such that a subset of its columns forms the identity matrix. This method returns the remaining part of the matrix.

For any given code, there can be many systematic generator matrices (depending on which positions should form the identity). This method will use the matrix returned by AbstractLinearCode.systematic_generator_matrix().

OUTPUT:

• An $$k \times (n-k)$$ matrix.

EXAMPLES:

sage: C = codes.HammingCode(GF(2), 3)
sage: C.generator_matrix()
[1 0 0 0 0 1 1]
[0 1 0 0 1 0 1]
[0 0 1 0 1 1 0]
[0 0 0 1 1 1 1]
sage: C.redundancy_matrix()
[0 1 1]
[1 0 1]
[1 1 0]
[1 1 1]
sage: C = LinearCode(matrix(GF(3),2,[1,2,0,\
2,1,1]))
sage: C.systematic_generator_matrix()
[1 2 0]
[0 0 1]
sage: C.redundancy_matrix()
[2]
[0]

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 non-negative integers: the weight distribution.

Warning

Specifying algorithm = "leon" sometimes prints a traceback related to a stack smashing error in the C library. The result appears to be computed correctly, however. It appears to run much faster than the GAP algorithm in small examples and much slower than the GAP algorithm in larger examples.

EXAMPLES:

sage: MS = MatrixSpace(GF(2),4,7)
sage: G = MS([[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])
sage: C = LinearCode(G)
sage: C.weight_distribution()
[1, 0, 0, 7, 7, 0, 0, 1]
sage: F.<z> = GF(2^2,"z")
sage: C = codes.HammingCode(F, 2); C
[5, 3] Hamming Code over GF(4)
sage: C.weight_distribution()
[1, 0, 0, 30, 15, 18]
sage: C = codes.HammingCode(GF(2), 3); C
[7, 4] Hamming Code over GF(2)
sage: C.weight_distribution(algorithm="leon")   # optional - gap_packages (Guava package)
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C.weight_distribution(algorithm="gap")
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C.weight_distribution(algorithm="binary")
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C = codes.HammingCode(GF(3), 3); C
[13, 10] Hamming Code over GF(3)
sage: C.weight_distribution() == C.weight_distribution(algorithm="leon")   # optional - gap_packages (Guava package)
True
sage: C = codes.HammingCode(GF(5), 2); C
[6, 4] Hamming Code over GF(5)
sage: C.weight_distribution() == C.weight_distribution(algorithm="leon")   # optional - gap_packages (Guava package)
True
sage: C = codes.HammingCode(GF(7), 2); C
[8, 6] Hamming Code over GF(7)
sage: C.weight_distribution() == C.weight_distribution(algorithm="leon")   # optional - gap_packages (Guava package)
True

standard_form(return_permutation=True)

Returns a linear code which is permutation-equivalent to self and admits a generator matrix in standard form.

A generator matrix is in standard form if it is of the form $$[I \vert A]$$, where $$I$$ is the $$k \times k$$ identity matrix. Any code admits a generator matrix in systematic form, i.e. where a subset of the columns form the identity matrix, but one might need to permute columns to allow the identity matrix to be leading.

INPUT:

• return_permutation – (default: True) if True, the column permutation which brings self into the returned code is also returned.

OUTPUT:

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 of self. If r belongs to self, its syndrome equals to the zero vector.

INPUT:

• r – a vector of the same length as self

OUTPUT:

• a column vector

EXAMPLES:

sage: MS = MatrixSpace(GF(2),4,7)
sage: G  = MS([[1,1,1,0,0,0,0], [1,0,0,1,1,0,0], [0,1,0,1,0,1,0], [1,1,0,1,0,0,1]])
sage: C  = LinearCode(G)
sage: r = vector(GF(2), (1,0,1,0,1,0,1))
sage: r in C
True
sage: C.syndrome(r)
(0, 0, 0)


If r is not a codeword, its syndrome is not equal to zero:

sage: r = vector(GF(2), (1,0,1,0,1,1,1))
sage: r in C
False
sage: C.syndrome(r)
(0, 1, 1)


Syndrome computation works fine on bigger fields:

sage: C = codes.random_linear_code(GF(59), 12, 4)
sage: c = C.random_element()
sage: C.syndrome(c)
(0, 0, 0, 0, 0, 0, 0, 0)

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 for LinearCodeSystematicEncoder details.

EXAMPLES:

sage: G = matrix(GF(3), [[ 1, 2, 1, 0],                                     [ 2, 1, 1, 1]])
sage: C = LinearCode(G)
sage: C.generator_matrix()
[1 2 1 0]
[2 1 1 1]
sage: C.systematic_generator_matrix()
[1 2 0 1]
[0 0 1 2]


Specific systematic positions can also be requested:

sage: C.systematic_generator_matrix(systematic_positions=[3,2]) [1 2 0 1] [1 2 1 0]
u_u_plus_v_code(other)

The $$(u|u+v)$$-construction with self=u and other=v

Returns the code obtained through $$(u|u+v)$$-construction with self as $$u$$ and other 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)

unencode(c, encoder_name=None, nocheck=False, **kwargs)

Returns the message corresponding to c.

This is the inverse of encode().

INPUT:

• c – a codeword of self.
• encoder_name – (default: None) name of the decoder which will be used to decode word. The default decoder of self will be used if default value is kept.
• nocheck – (default: False) checks if c is in self. You might set this to True to disable the check for saving computation. Note that if c is not in self and nocheck = True, then the output of unencode() is not defined (except that it will be in the message space of self).
• kwargs – all additional arguments are forwarded to the construction of the encoder that is used.

OUTPUT:

• an element of the message space of encoder_name of self.

EXAMPLES:

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])
sage: C = LinearCode(G)
sage: c = vector(GF(2), (1, 1, 0, 0, 1, 1, 0))
sage: C.unencode(c)
(0, 1, 1, 0)

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 non-negative integers: the weight distribution.

Warning

Specifying algorithm = "leon" sometimes prints a traceback related to a stack smashing error in the C library. The result appears to be computed correctly, however. It appears to run much faster than the GAP algorithm in small examples and much slower than the GAP algorithm in larger examples.

EXAMPLES:

sage: MS = MatrixSpace(GF(2),4,7)
sage: G = MS([[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])
sage: C = LinearCode(G)
sage: C.weight_distribution()
[1, 0, 0, 7, 7, 0, 0, 1]
sage: F.<z> = GF(2^2,"z")
sage: C = codes.HammingCode(F, 2); C
[5, 3] Hamming Code over GF(4)
sage: C.weight_distribution()
[1, 0, 0, 30, 15, 18]
sage: C = codes.HammingCode(GF(2), 3); C
[7, 4] Hamming Code over GF(2)
sage: C.weight_distribution(algorithm="leon")   # optional - gap_packages (Guava package)
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C.weight_distribution(algorithm="gap")
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C.weight_distribution(algorithm="binary")
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C = codes.HammingCode(GF(3), 3); C
[13, 10] Hamming Code over GF(3)
sage: C.weight_distribution() == C.weight_distribution(algorithm="leon")   # optional - gap_packages (Guava package)
True
sage: C = codes.HammingCode(GF(5), 2); C
[6, 4] Hamming Code over GF(5)
sage: C.weight_distribution() == C.weight_distribution(algorithm="leon")   # optional - gap_packages (Guava package)
True
sage: C = codes.HammingCode(GF(7), 2); C
[8, 6] Hamming Code over GF(7)
sage: C.weight_distribution() == C.weight_distribution(algorithm="leon")   # optional - gap_packages (Guava package)
True

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 $$x^i y^{n-i}$$ is the number of codewords of $$self$$ of Hamming weight $$i$$. Here, $$n$$ is the length of $$self$$.

INPUT:

• names - (default: "xy") The names of the variables in the homogeneous polynomial. Can be given as a single string of length 2, or a single string with a comma, or as a tuple or list of two strings.
• bivariate - (default: $$True$$) Whether to return a bivariate, homogeneous polynomial or just a univariate polynomial. If set to False, then names will be interpreted as a single variable name and default to "x".

OUTPUT:

• The weight enumerator polynomial over $$\ZZ$$.

EXAMPLES:

sage: C = codes.HammingCode(GF(2), 3)
sage: C.weight_enumerator()
x^7 + 7*x^4*y^3 + 7*x^3*y^4 + y^7
sage: C.weight_enumerator(names="st")
s^7 + 7*s^4*t^3 + 7*s^3*t^4 + t^7
sage: C.weight_enumerator(names="var1, var2")
var1^7 + 7*var1^4*var2^3 + 7*var1^3*var2^4 + var2^7
sage: C.weight_enumerator(names=('var1', 'var2'))
var1^7 + 7*var1^4*var2^3 + 7*var1^3*var2^4 + var2^7
sage: C.weight_enumerator(bivariate=False)
x^7 + 7*x^4 + 7*x^3 + 1


An example of a code with a non-symmetrical weight enumerator:

sage: C = codes.GolayCode(GF(3), extended=False)
sage: C.weight_enumerator()
24*x^11 + 110*x^9*y^2 + 330*x^8*y^3 + 132*x^6*y^5 + 132*x^5*y^6 + y^11

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)

Linear codes over a finite field or finite ring, represented using a generator matrix.

This class should be used for arbitrary and unstructured linear codes. This means that basic operations on the code, such as the computation of the minimum distance, will use generic, slow algorithms.

If you are looking for constructing a code from a more specific family, see if the family has been implemented by investigating $$codes.<tab>$$. These more specific classes use properties particular to that family to allow faster algorithms, and could also have family-specific methods.

INPUT:

• generator – a generator matrix over a finite field (G can be defined over a finite ring but the matrices over that ring must have certain attributes, such as rank); or a code over a finite field
• d – (optional, default: None) the minimum distance of the code

Note

The veracity of the minimum distance d, if provided, is not checked.

EXAMPLES:

sage: MS = MatrixSpace(GF(2),4,7)
sage: G  = MS([[1,1,1,0,0,0,0], [1,0,0,1,1,0,0], [0,1,0,1,0,1,0], [1,1,0,1,0,0,1]])
sage: C  = LinearCode(G)
sage: C
[7, 4] linear code over GF(2)
sage: C.base_ring()
Finite Field of size 2
sage: C.dimension()
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 (11-2005)
• Charles Prior (03-2016): 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)

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:

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)

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 of self

OUTPUT:

• a vector of self’s message space

EXAMPLES:

sage: G = Matrix(GF(2), [[1,1,1,0,0,0,0],[1,0,0,1,1,0,0],[0,1,0,1,0,1,0],[1,1,0,1,0,0,1]])
sage: C = LinearCode(G)
sage: D = codes.decoders.LinearCodeNearestNeighborDecoder(C)
sage: word = vector(GF(2), (1, 1, 0, 0, 1, 1, 0))
sage: w_err = word + vector(GF(2), (1, 0, 0, 0, 0, 0, 0))
sage: D.decode_to_code(w_err)
(1, 1, 0, 0, 1, 1, 0)

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)
1

class sage.coding.linear_code.LinearCodeSyndromeDecoder(code, maximum_error_weight=None)

Constructs a decoder for Linear Codes based on syndrome lookup table.

The decoding algorithm works as follows:

• First, a lookup table is built by computing the syndrome of every error pattern of weight up to maximum_error_weight.
• Then, whenever one tries to decode a word r, the syndrome of r is computed. The corresponding error pattern is recovered from the pre-computed lookup table.
• Finally, the recovered error pattern is subtracted from r to recover the original word.

maximum_error_weight need never exceed the covering radius of the code, since there are then always lower-weight errors with the same syndrome. If one sets maximum_error_weight to a value greater than the covering radius, then the covering radius will be determined while building the lookup-table. This lower value is then returned if you query decoding_radius after construction.

If maximum_error_weight is left unspecified or set to a number at least the covering radius of the code, this decoder is complete, i.e. it decodes every vector in the ambient space.

Note

Constructing the lookup table takes time exponential in the length of the code and the size of the code’s base field. Afterwards, the individual decodings are fast.

INPUT:

• code – A code associated to this decoder
• maximum_error_weight – (default: None) the maximum number of errors to look for when building the table. An error is raised if it is set greater than $$n-k$$, since this is an upper bound on the covering radius on any linear code. If maximum_error_weight is kept unspecified, it will be set to $$n - k$$, where $$n$$ is the length of code and $$k$$ its dimension.

EXAMPLES:

sage: G = Matrix(GF(3), [[1,0,0,1,0,1,0,1,2],[0,1,0,2,2,0,1,1,0],[0,0,1,0,2,2,2,1,2]])
sage: C = LinearCode(G)
sage: D = codes.decoders.LinearCodeSyndromeDecoder(C)
sage: D
Syndrome decoder for [9, 3] linear code over GF(3) handling errors of weight up to 4


If one wants to correct up to a lower number of errors, one can do as follows:

sage: D = codes.decoders.LinearCodeSyndromeDecoder(C, maximum_error_weight=2)
sage: D
Syndrome decoder for [9, 3] linear code over GF(3) handling errors of weight up to 2


If one checks the list of types of this decoder before constructing it, one will notice it contains the keyword dynamic. Indeed, the behaviour of the syndrome decoder depends on the maximum error weight one wants to handle, and how it compares to the minimum distance and the covering radius of code. In the following examples, we illustrate this property by computing different instances of syndrome decoder for the same code.

We choose the following linear code, whose covering radius equals to 4 and minimum distance to 5 (half the minimum distance is 2):

sage: G = matrix(GF(5), [[1, 0, 0, 0, 0, 4, 3, 0, 3, 1, 0],
....:                    [0, 1, 0, 0, 0, 3, 2, 2, 3, 2, 1],
....:                    [0, 0, 1, 0, 0, 1, 3, 0, 1, 4, 1],
....:                    [0, 0, 0, 1, 0, 3, 4, 2, 2, 3, 3],
....:                    [0, 0, 0, 0, 1, 4, 2, 3, 2, 2, 1]])
sage: C = LinearCode(G)


In the following examples, we illustrate how the choice of maximum_error_weight influences the types of the instance of syndrome decoder, alongside with its decoding radius.

We build a first syndrome decoder, and pick a maximum_error_weight smaller than both the covering radius and half the minimum distance:

sage: D = C.decoder("Syndrome", maximum_error_weight = 1)
sage: D.decoder_type()
{'always-succeed', 'bounded_distance', 'hard-decision'}
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', 'hard-decision', 'might-error'}
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', 'hard-decision', 'might-error'}
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 of self

OUTPUT:

• a vector of self’s message space

EXAMPLES:

sage: G = Matrix(GF(3),[
....:   [1, 0, 0, 0, 2, 2, 1, 1],
....:   [0, 1, 0, 0, 0, 0, 1, 1],
....:   [0, 0, 1, 0, 2, 0, 0, 2],
....:   [0, 0, 0, 1, 0, 2, 0, 1]])
sage: C = LinearCode(G)
sage: D = codes.decoders.LinearCodeSyndromeDecoder(C, maximum_error_weight = 2)
sage: Chan = channels.StaticErrorRateChannel(C.ambient_space(), 2)
sage: c = C.random_element()
sage: r = Chan(c)
sage: c == D.decode_to_code(r)
True

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)
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)}

class sage.coding.linear_code.LinearCodeSystematicEncoder(code, systematic_positions=None)

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 $$n-1$$ where $$n$$ is the length of the code and $$k$$ its dimension. The 0th symbol of a message will then be at position systematic_positions[0], the 1st index at position systematic_positions[1], etc. A ValueError is raised at construction time if the supplied indices do not form an information set.

EXAMPLES:

The following demonstrates the basic usage of 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"
....:
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,..,k-1]

EXAMPLES:

sage: C = LinearCode(matrix(GF(2), [[1,0,0,0,1,1,0],\
[0,1,0,1,0,1,0],\
[0,0,0,0,0,0,1]]))
sage: E = codes.encoders.LinearCodeSystematicEncoder(C)
sage: E.systematic_positions()
(0, 1, 6)
sage: E.systematic_permutation()
[1, 2, 7, 3, 4, 5, 6]

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)