# Linear Codes¶

VERSION: 1.2

Let $$F$$ be a finite field. Here, we will denote the finite field with $$q$$ elements by $$\GF{q}$$. A subspace of $$F^n$$ (with the standard basis) is called a linear code of length $$n$$. If its dimension is denoted $$k$$ then we typically store a basis of $$C$$ as a $$k \times n$$ matrix, with rows the basis vectors. It is called the generator matrix of $$C$$. The rows of the parity check matrix of $$C$$ are a basis for the code,

$C^* = \{ v \in GF(q)^n\ |\ v\cdot c = 0,\ for \ all\ c \in C \},$

called the dual space of $$C$$.

If $$F=\GF{2}$$ then $$C$$ is called a binary code. If $$F = \GF{q}$$ then $$C$$ is called a $$q$$-ary code. The elements of a code $$C$$ are called codewords.

Let $$C$$, $$D$$ be linear codes of length $$n$$ and dimension $$k$$. There are several notions of equivalence for linear codes:

$$C$$ and $$D$$ are

• permutational equivalent, if there is some permutation $$\pi \in S_n$$ such that $$(c_{\pi(0)}, \ldots, c_{\pi(n-1)}) \in D$$ for all $$c \in C$$.
• linear equivalent, if there is some permutation $$\pi \in S_n$$ and a vector $$\phi$$ of units of length $$n$$ such that $$(c_{\pi(0)} \phi_0^{-1}, \ldots, c_{\pi(n-1)} \phi_{n-1}^{-1}) \in D$$ for all $$c \in C$$.
• semilinear equivalent, if there is some permutation $$\pi \in S_n$$, a vector $$\phi$$ of units of length $$n$$ and a field automorphism $$\alpha$$ such that $$(\alpha(c_{\pi(0)}) \phi_0^{-1}, \ldots, \alpha( c_{\pi(n-1)}) \phi_{n-1}^{-1} ) \in D$$ for all $$c \in C$$.

These are group actions. If one of these group elements sends the linear code $$C$$ to itself, then we will call it an automorphism. Depending on the group action we will call those groups:

• permuation automorphism group
• monomial automorphism group (every linear Hamming isometry is a monomial transformation of the ambient space, for $$n\geq 3$$)
• automorphism group (every semilinear Hamming isometry is a semimonomial transformation of the ambient space, for $$n\geq 3$$)

This file contains

1. LinearCode class definition; LinearCodeFromVectorspace conversion function,
2. The spectrum (weight distribution), covering_radius, minimum distance programs (calling Steve Linton’s or CJ Tjhal’s C programs), characteristic_function, and several implementations of the Duursma zeta function (sd_zeta_polynomial, zeta_polynomial, zeta_function, chinen_polynomial, for example),
3. interface with best_known_linear_code_www (interface with codetables.de since A. Brouwer’s online tables have been disabled), bounds_minimum_distance which call tables in GUAVA (updated May 2006) created by Cen Tjhai instead of the online internet tables,
4. gen_mat, gen_mat_systematic, information_set, list, check_mat, decode, dual_code, extended_code, shortened, punctured, genus, binomial_moment, and divisor methods for LinearCode,
5. Boolean-valued functions such as “==”, is_self_dual, is_self_orthogonal, is_subcode, is_permutation_automorphism, is_permutation_equivalent (which interfaces with Robert Miller’s partition refinement code),
6. permutation methods: is_permutation_automorphism, permutation_automorphism_group, permuted_code, standard_form, module_composition_factors,
7. design-theoretic methods: assmus_mattson_designs (implementing Assmus-Mattson Theorem),
8. code constructions, such as HammingCode and ToricCode, are in a separate code_constructions.py module; in the separate guava.py module, you will find constructions, such as RandomLinearCodeGuava and BinaryReedMullerCode, wrapped from the corresponding GUAVA codes.

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.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


1. More wrappers
2. GRS codes and special decoders.
3. $$P^1$$ Goppa codes and group actions on $$P^1$$ RR space codes.

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 gen_mat_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

TESTS:

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)
True

class sage.coding.linear_code.LinearCode(gen_mat, d=None)

Linear codes over a finite field or finite ring.

A linear code is a subspace of a vector space over a finite field. It can be defined by one of its basis or equivalently a generator matrix (a $$k \times n$$ matrix of full rank $$k$$).

INPUT:

• gen_mat – 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)
• 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
Linear code of length 7, dimension 4 over Finite Field of size 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
Linear code of length 7, dimension 4 over Finite Field of size 5


AUTHORS:

• David Joyner (11-2005)
ambient_space()

Returns the ambient vector space of $$self$$.

EXAMPLES:

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

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

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 [HP])


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.ExtendedBinaryGolayCode()             #  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


REFERENCE:

• [HP] W. C. Huffman and V. Pless, Fundamentals of ECC, Cambridge Univ. Press, 2003.
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:

sage: C = codes.HammingCode(3,GF(4,"z"));
sage: C.automorphism_group_gens()
([((1, 1, 1, z, z + 1, z + 1, z + 1, z, z, 1, 1, 1, z, z, z + 1, z, z, z + 1, z + 1, z + 1, 1); (1,6,12,17)(2,16,4,5,11,8,14,13)(3,21,19,10,20,18,15,9), Ring endomorphism of Finite Field in z of size 2^2
Defn: z |--> z + 1), ((1, 1, 1, z, z + 1, 1, 1, z, z, z + 1, z, z, z + 1, z + 1, z + 1, 1, z + 1, z, z, 1, 1); (1,6,9,13,15,18)(2,21)(3,16,7)(4,5,11,10,12,14)(17,19), Ring endomorphism of Finite Field in z of size 2^2
Defn: z |--> z), ((z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z); (), Ring endomorphism of Finite Field in z of size 2^2
Defn: z |--> z)], 362880)
sage: C.automorphism_group_gens(equivalence="linear")
([((z, z, 1, 1, z + 1, z, z + 1, z, z, z + 1, 1, 1, 1, z + 1, z, z, z + 1, z + 1, 1, 1, z); (1,5,10,9,4,14,11,16,18,20,6,19,12,15,3,8,2,17,7,13,21), Ring endomorphism of Finite Field in z of size 2^2
Defn: z |--> z), ((z + 1, 1, z, 1, 1, z + 1, z + 1, z, 1, z, z + 1, z, z + 1, z + 1, z, 1, 1, z + 1, z + 1, z + 1, z); (1,17,10)(2,15,13)(4,11,21)(5,18,12)(6,14,19)(7,8,16), Ring endomorphism of Finite Field in z of size 2^2
Defn: z |--> z), ((z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1, z + 1); (), Ring endomorphism of Finite Field in z of size 2^2
Defn: z |--> z)], 181440)
sage: C.automorphism_group_gens(equivalence="permutational")
([((1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1); (1,11)(3,10)(4,9)(5,7)(12,21)(14,20)(15,19)(16,17), Ring endomorphism of Finite Field in z of size 2^2
Defn: z |--> z), ((1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1); (2,18)(3,19)(4,10)(5,16)(8,13)(9,14)(11,21)(15,20), Ring endomorphism of Finite Field in z of size 2^2
Defn: z |--> z), ((1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1); (1,19)(3,17)(4,21)(5,20)(7,14)(9,12)(10,16)(11,15), Ring endomorphism of Finite Field in z of size 2^2
Defn: z |--> z), ((1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1); (2,13)(3,14)(4,20)(5,11)(8,18)(9,19)(10,15)(16,21), Ring endomorphism of Finite Field in z of size 2^2
Defn: z |--> z)], 64)

basis()

Returns a basis of $$self$$.

EXAMPLES:

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

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(3,GF(2))
sage: C.binomial_moment(2)
0
sage: C.binomial_moment(4)    # long time
35


Warning

This is slow.

REFERENCE:

• I. Duursma, “Combinatorics of the two-variable zeta function”, Finite fields and applications, 109-136, Lecture Notes in Comput. Sci., 2948, Springer, Berlin, 2004.
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(3,F)
sage: CanRep, transp = C.canonical_representative()


Check that the transporter element is correct:

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


Check if an equivalent code has the same canonical representative:

sage: f = F.hom([z**2])
sage: C_iso = LinearCode(C.gen_mat().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(3, GF(2))
sage: C.cardinality()
16
sage: len(C)
16

characteristic()

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

EXAMPLES:

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

characteristic_polynomial()

Returns the characteristic polynomial of a linear code, as defined in van Lint’s text [vL].

EXAMPLES:

sage: C = codes.ExtendedBinaryGolayCode()
sage: C.characteristic_polynomial()
-4/3*x^3 + 64*x^2 - 2816/3*x + 4096


REFERENCES:

• van Lint, Introduction to coding theory, 3rd ed., Springer-Verlag GTM, 86, 1999.
check_mat()

Returns the check matrix of self.

EXAMPLES:

sage: C = codes.HammingCode(3,GF(2))
sage: Cperp = C.dual_code()
sage: C; Cperp
Linear code of length 7, dimension 4 over Finite Field of size 2
Linear code of length 7, dimension 3 over Finite Field of size 2
sage: C.gen_mat()
[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.check_mat()
[1 0 1 0 1 0 1]
[0 1 1 0 0 1 1]
[0 0 0 1 1 1 1]
sage: Cperp.check_mat()
[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.gen_mat()
[1 0 1 0 1 0 1]
[0 1 1 0 0 1 1]
[0 0 0 1 1 1 1]

chinen_polynomial()

Returns the Chinen zeta polynomial of the code.

EXAMPLES:

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

The covering radius of a linear code $$C$$ is the smallest number $$r$$ with the property that each element $$v$$ of the ambient vector space of $$C$$ has at most a distance $$r$$ to the code $$C$$. So for each vector $$v$$ there must be an element $$c$$ of $$C$$ with $$d(v,c) \leq r$$. A binary linear code with reasonable small covering radius is often referred to as a covering code.

For example, if $$C$$ is a perfect code, the covering radius is equal to $$t$$, the number of errors the code can correct, where $$d = 2t+1$$, with $$d$$ the minimum distance of $$C$$.

EXAMPLES:

sage: C = codes.HammingCode(5,GF(2))
sage: C.covering_radius()  # optional - gap_packages (Guava package)
1

decode(right, algorithm='syndrome')

Decodes the received vector right to an element $$c$$ in this code.

Optional algorithms are “guava”, “nearest neighbor” or “syndrome”. The algorithm="guava" wraps GUAVA’s Decodeword. Hamming codes have a special decoding algorithm; otherwise, "syndrome" decoding is used.

INPUT:

• right - Vector of length the length of this code
• algorithm - Algorithm to use, one of "syndrome", "nearest neighbor", and "guava" (default: "syndrome")

OUTPUT:

• The codeword in this code closest to right.

EXAMPLES:

sage: C = codes.HammingCode(3,GF(2))
sage: MS = MatrixSpace(GF(2),1,7)
sage: F = GF(2); a = F.gen()
sage: v1 = [a,a,F(0),a,a,F(0),a]
sage: C.decode(v1)
(1, 1, 0, 1, 0, 0, 1)
sage: C.decode(v1,algorithm="nearest neighbor")
(1, 1, 0, 1, 0, 0, 1)
sage: C.decode(v1,algorithm="guava")  # optional - gap_packages (Guava package)
(1, 1, 0, 1, 0, 0, 1)
sage: v2 = matrix([[a,a,F(0),a,a,F(0),a]])
sage: C.decode(v2)
(1, 1, 0, 1, 0, 0, 1)
sage: v3 = vector([a,a,F(0),a,a,F(0),a])
sage: c = C.decode(v3); c
(1, 1, 0, 1, 0, 0, 1)
sage: c in C
True
sage: C = codes.HammingCode(2,GF(5))
sage: v = vector(GF(5),[1,0,0,2,1,0])
sage: C.decode(v)
(1, 0, 0, 2, 2, 0)
sage: F.<a> = GF(4)
sage: C = codes.HammingCode(2,F)
sage: v = vector(F, [1,0,0,a,1])
sage: C.decode(v)
(a + 1, 0, 0, a, 1)
sage: C.decode(v, algorithm="nearest neighbor")
(a + 1, 0, 0, a, 1)
sage: C.decode(v, algorithm="guava")  # optional - gap_packages (Guava package)
(a + 1, 0, 0, a, 1)


Does not work for very long codes since the syndrome table grows too large.

TESTS:

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

sage: (C.decode(v)).is_immutable()
True

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)

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(3,GF(2))
sage: C2 = C1.direct_sum(C1); C2
Linear code of length 14, dimension 8 over Finite Field of size 2
sage: C3 = C1.direct_sum(C2); C3
Linear code of length 21, dimension 12 over Finite Field of size 2

divisor()

Returns the divisor of a code, which is the smallest integer $$d_0 > 0$$ such that each $$A_i > 0$$ iff $$i$$ is divisible by $$d_0$$.

EXAMPLES:

sage: C = codes.ExtendedBinaryGolayCode()
sage: C.divisor()   # Type II self-dual
4
sage: C.divisor()
2

dual_code()

This computes the dual code $$Cd$$ of the code $$C$$,

$Cd = \{ v \in V\ |\ v\cdot c = 0,\ \forall c \in C \}.$

Does not call GAP.

EXAMPLES:

sage: C = codes.HammingCode(3,GF(2))
sage: C.dual_code()
Linear code of length 7, dimension 3 over Finite Field of size 2
sage: C = codes.HammingCode(3,GF(4,'a'))
sage: C.dual_code()
Linear code of length 21, dimension 3 over Finite Field in a of size 2^2

extended_code()

If self is a linear code of length $$n$$ defined over $$F$$ then this returns the code of length $$n+1$$ where the last digit $$c_n$$ satisfies the check condition $$c_0+...+c_n=0$$. If self is an $$[n,k,d]$$ binary code then the extended code $$C^{\vee}$$ is an $$[n+1,k,d^{\vee}]$$ code, where $$d^=d$$ (if d is even) and $$d^{\vee}=d+1$$ (if $$d$$ is odd).

EXAMPLES:

sage: C = codes.HammingCode(3,GF(4,'a'))
sage: C
Linear code of length 21, dimension 18 over Finite Field in a of size 2^2
sage: Cx = C.extended_code()
sage: Cx
Linear code of length 22, dimension 18 over Finite Field in a of size 2^2

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(3,GF(4,'a'))
sage: Cc = C.galois_closure(GF(2))
sage: C; Cc
Linear code of length 21, dimension 18 over Finite Field in a of size 2^2
Linear code of length 21, dimension 20 over Finite Field in a of size 2^2
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

gen_mat()

Return a generator matrix of this code.

EXAMPLES:

sage: C1 = codes.HammingCode(3,GF(2))
sage: C1.gen_mat()
[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: C2 = codes.HammingCode(2,GF(4,"a"))
sage: C2.gen_mat()
[    1     0     0 a + 1     a]
[    0     1     0     1     1]
[    0     0     1     a a + 1]

gen_mat_systematic()

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.

EXAMPLES:

sage: G = matrix(GF(3),2,[1,-1,1,-1,1,1])
sage: code = LinearCode(G)
sage: code.gen_mat()
[1 2 1]
[2 1 1]
sage: code.gen_mat_systematic()
[1 2 0]
[0 0 1]

gens()

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

EXAMPLES:

sage: C = codes.HammingCode(3,GF(2))
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(3,GF(2)); C1
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: C1.genus()
1
sage: C2 = codes.HammingCode(2,GF(4,"a")); C2
Linear code of length 5, dimension 3 over Finite Field in a of size 2^2
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.

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,-1,0,-1,1,1])
sage: code = LinearCode(G)
sage: code.gen_mat_systematic()
[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(3,GF(4,"a"))
sage: C.is_galois_closed()
False

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(3,GF(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.CyclicCodeFromGeneratingPolynomial(7,g); C1
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: C2 = codes.HammingCode(3,GF(2)); C2
Linear code of length 7, dimension 4 over Finite Field of size 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.RandomLinearCode(10,5,GF(2))
sage: C2 = codes.RandomLinearCode(10,5,GF(3))
sage: C1.is_permutation_equivalent(C2)
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.ExtendedBinaryGolayCode()
sage: C.is_self_dual()
True
sage: C = codes.HammingCode(3,GF(2))
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.ExtendedBinaryGolayCode()
sage: C.is_self_orthogonal()
True
sage: C = codes.HammingCode(3,GF(2))
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(3,GF(2))
sage: G1 = C1.gen_mat()
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(3,GF(9,"z"))
sage: G1 = C1.gen_mat()
sage: G2 = G1.matrix_from_rows([0,1,2])
sage: C2 = LinearCode(G2)
sage: C2.is_subcode(C1)
True

length()

Returns the length of this code.

EXAMPLES:

sage: C = codes.HammingCode(3,GF(2))
sage: C.length()
7

list()

Return a list of all elements of this linear code.

EXAMPLES:

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

By default, this uses a GAP kernel function (in C and not part of Guava) written by Steve Linton. If algorithm="guava" is set and $$q$$ is 2 or 3 then this uses a very fast program written in C written by CJ Tjhal. (This is much faster, except in some small examples.)

Raises a ValueError in case there is no non-zero vector in this linear code.

The minimum distance of the code is stored once it has been computed or provided during the initialization of LinearCode. If algorithm is None and the stored value of minimum distance is found, then the stored value will be returned without recomputing the minimum distance again.

INPUT:

• algorithm - Method to be used, None, "gap", or "guava" (default: None).

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


Once the minimum distance has been computed, it’s value is stored. Hence the following command will return the value instantly, without further computations.:

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


Another example.:

sage: C = codes.HammingCode(2,GF(4,"a")); C
Linear code of length 5, dimension 3 over Finite Field in a of size 2^2
sage: C.minimum_distance()
3


TESTS:

sage: C = codes.HammingCode(2,GF(4,"a"))
sage: C.minimum_distance(algorithm='something')
Traceback (most recent call last):
...
ValueError: The algorithm argument must be one of None, 'gap' or 'guava'; got 'something'

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.

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
Linear code of length 8, dimension 4 over Finite Field of size 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.ExtendedTernaryGolayCode()
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.ExtendedBinaryGolayCode()
sage: G = C.permutation_automorphism_group()
sage: G.order()
244823040
sage: C = codes.HammingCode(5, GF(2))
sage: G = C.permutation_automorphism_group()
sage: G.order()
9999360
sage: C = codes.HammingCode(2,GF(3)); C
Linear code of length 4, dimension 2 over Finite Field of size 3
sage: C.permutation_automorphism_group(algorithm="partition")
Permutation Group with generators [(1,3,4)]
sage: C = codes.HammingCode(2,GF(4,"z")); C
Linear code of length 5, dimension 3 over Finite Field in z of size 2^2
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)
sage: C = codes.TernaryGolayCode()
sage: C.permutation_automorphism_group(algorithm="gap")  # optional - gap_packages (Guava package)
Permutation Group with generators [(3,4)(5,7)(6,9)(8,11), (3,5,8)(4,11,7)(6,9,10), (2,3)(4,6)(5,8)(7,10), (1,2)(4,11)(5,8)(9,10)]


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(3,GF(2))
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
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: C == Cg
True

punctured(L)

Returns the code punctured at the positions $$L$$, $$L \subset \{1,2,...,n\}$$. If this code $$C$$ is of length $$n$$ in GF(q) then the code $$C^L$$ obtained from $$C$$ by puncturing at the positions in $$L$$ is the code of length $$n-L$$ consisting of codewords of $$C$$ which have their $$i-th$$ coordinate deleted if $$i \in L$$ and left alone if $$i\notin L$$:

$C^L = \{(c_{i_1},...,c_{i_N})\ |\ (c_1,...,c_n)\in C\},$

where $$\{1,2,...,n\}-T = \{i_1,...,i_N\}$$. In particular, if $$L=\{j\}$$ then $$C^L$$ is simply the code obtainen from $$C$$ by deleting the $$j-th$$ coordinate of each codeword. The code $$C^L$$ is called the punctured code at $$L$$. The dimension of $$C^L$$ can decrease if $$|L|>d-1$$.

INPUT:

• L - Subset of $$\{1,...,n\}$$, where $$n$$ is the length of self

OUTPUT:

• Linear code, the punctured code described above

EXAMPLES:

sage: C = codes.HammingCode(3,GF(2))
sage: C.punctured([1,2])
Linear code of length 5, dimension 4 over Finite Field of size 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(3,GF(4,'a'))
sage: C.random_element() # random test
(1, 0, 0, a + 1, 1, a, a, a + 1, a + 1, 1, 1, 0, a + 1, a, 0, a, a, 0, a, a, 1)


Passes extra positional or keyword arguments through:

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


TESTS:

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

sage: c = C.random_element()
sage: c.is_immutable()
True

redundancy_matrix(C)

If C is a linear [n,k,d] code then this function returns a $$k imes (n-k)$$ matrix A such that G = (I,A) generates a code (in standard form) equivalent to C. If C is already in standard form and G = (I,A) is its generator matrix then this function simply returns that A.

OUTPUT:

• Matrix, the redundancy matrix

EXAMPLES:

sage: C = codes.HammingCode(3,GF(2))
sage: C.gen_mat()
[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.standard_form()[0].gen_mat()
[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 = codes.HammingCode(2,GF(3))
sage: C.gen_mat()
[1 0 1 1]
[0 1 1 2]
sage: C.redundancy_matrix()
[1 1]
[1 2]

sd_duursma_data(C, i)

Returns the Duursma data $$v$$ and $$m$$ of this formally s.d. code $$C$$ and the type number $$i$$ in (1,2,3,4). Does not check if this code is actually sd.

INPUT:

• i - Type number

OUTPUT:

• Pair (v, m) as in Duursma [D]

REFERENCES:

 [D] (1, 2, 3) I. Duursma, “Extremal weight enumerators and ultraspherical polynomials”

EXAMPLES:

sage: MS = MatrixSpace(GF(2),2,4)
sage: G = MS([1,1,0,0,0,0,1,1])
sage: C = LinearCode(G)
sage: C == C.dual_code()  # checks that C is self dual
True
sage: for i in [1,2,3,4]: print C.sd_duursma_data(i)
[2, -1]
[2, -3]
[2, -2]
[2, -1]

sd_duursma_q(C, i, d0)

INPUT:

• C - sd code; does not check if $$C$$ is actually an sd code
• i - Type number, one of 1,2,3,4
• d0 - Divisor, the smallest integer such that each $$A_i > 0$$ iff $$i$$ is divisible by $$d0$$

OUTPUT:

• Coefficients $$q_0, q_1, ...$$ of $$q(T)$$ as in Duursma [D]

REFERENCES:

• [D] - I. Duursma, “Extremal weight enumerators and ultraspherical polynomials”

EXAMPLES:

sage: C1 = codes.HammingCode(3,GF(2))
sage: C2 = C1.extended_code(); C2
Linear code of length 8, dimension 4 over Finite Field of size 2
sage: C2.is_self_dual()
True
sage: C2.sd_duursma_q(1,1)
2/5*T^2 + 2/5*T + 1/5
sage: C2.sd_duursma_q(3,1)
3/5*T^4 + 1/5*T^3 + 1/15*T^2 + 1/15*T + 1/15

sd_zeta_polynomial(C, typ=1)

Returns the Duursma zeta function of a self-dual code using the construction in [D].

INPUT:

• typ - Integer, type of this s.d. code; one of 1,2,3, or 4 (default: 1)

OUTPUT:

• Polynomial

EXAMPLES:

sage: C1 = codes.HammingCode(3,GF(2))
sage: C2 = C1.extended_code(); C2
Linear code of length 8, dimension 4 over Finite Field of size 2
sage: C2.is_self_dual()
True
sage: C2.sd_zeta_polynomial()
2/5*T^2 + 2/5*T + 1/5
sage: C2.zeta_polynomial()
2/5*T^2 + 2/5*T + 1/5
sage: P = C2.sd_zeta_polynomial(); P(1)
1
sage: F.<z> = GF(4,"z")
sage: MS = MatrixSpace(F, 3, 6)
sage: G = MS([[1,0,0,1,z,z],[0,1,0,z,1,z],[0,0,1,z,z,1]])
sage: C = LinearCode(G)  # the "hexacode"
sage: C.sd_zeta_polynomial(4)
1


It is a general fact about Duursma zeta polynomials that $$P(1) = 1$$.

REFERENCES:

• [D] I. Duursma, “Extremal weight enumerators and ultraspherical polynomials”
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 [HP], $$C_L$$ 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(3,GF(2))
sage: C.shortened([1,2])
Linear code of length 5, dimension 2 over Finite Field of size 2

spectrum(algorithm=None)

Returns the spectrum of self as a list.

The default algorithm uses a GAP kernel function (in C) written by Steve Linton.

INPUT:

• algorithm - None, "gap", "leon", or "binary"; defaults to "gap" except in the binary case. If "gap" then uses the GAP function, if "leon" then uses Jeffrey Leon’s software via Guava, and if "binary" then uses Sage native Cython code
• List, the spectrum

The optional algorithm ("leon") may create a stack smashing error and a traceback but should return the correct answer. It appears to run much faster than the GAP algorithm in some small examples and much slower than the GAP algorithm in other 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.spectrum()
[1, 0, 0, 7, 7, 0, 0, 1]
sage: F.<z> = GF(2^2,"z")
sage: C = codes.HammingCode(2, F); C
Linear code of length 5, dimension 3 over Finite Field in z of size 2^2
sage: C.spectrum()
[1, 0, 0, 30, 15, 18]
sage: C = codes.HammingCode(3,GF(2)); C
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: C.spectrum(algorithm="leon")   # optional - gap_packages (Guava package)
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C.spectrum(algorithm="gap")
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C.spectrum(algorithm="binary")
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C = codes.HammingCode(3,GF(3)); C
Linear code of length 13, dimension 10 over Finite Field of size 3
sage: C.spectrum() == C.spectrum(algorithm="leon")   # optional - gap_packages (Guava package)
True
sage: C = codes.HammingCode(2,GF(5)); C
Linear code of length 6, dimension 4 over Finite Field of size 5
sage: C.spectrum() == C.spectrum(algorithm="leon")   # optional - gap_packages (Guava package)
True
sage: C = codes.HammingCode(2,GF(7)); C
Linear code of length 8, dimension 6 over Finite Field of size 7
sage: C.spectrum() == C.spectrum(algorithm="leon")   # optional - gap_packages (Guava package)
True

standard_form()

Returns the standard form of this linear code.

An $$[n,k]$$ linear code with generator matrix $$G$$ in standard form is the row-reduced echelon form of $$G$$ is $$(I,A)$$, where $$I$$ denotes the $$k \times k$$ identity matrix and $$A$$ is a $$k \times (n-k)$$ block. This method returns a pair $$(C,p)$$ where $$C$$ is a code permutation equivalent to self and $$p$$ in $$S_n$$, with $$n$$ the length of $$C$$, is the permutation sending self to $$C$$. This does not call GAP.

Thanks to Frank Luebeck for (the GAP version of) this code.

EXAMPLES:

sage: C = codes.HammingCode(3,GF(2))
sage: C.gen_mat()
[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: MS = MatrixSpace(GF(3),3,7)
sage: G = MS([[1,0,0,0,1,1,0],[0,1,0,1,0,1,0],[0,0,0,0,0,0,1]])
sage: C = LinearCode(G)
sage: Cs, p = C.standard_form()
sage: p
(3,7)
sage: Cs.gen_mat()
[1 0 0 0 1 1 0]
[0 1 0 1 0 1 0]
[0 0 1 0 0 0 0]

support()

Returns the set of indices $$j$$ where $$A_j$$ is nonzero, where spectrum(self) = $$[A_0,A_1,...,A_n]$$.

OUTPUT:

• List of integers

EXAMPLES:

sage: C = codes.HammingCode(3,GF(2))
sage: C.spectrum()
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C.support()
[0, 3, 4, 7]

weight_distribution(algorithm=None)

Returns the spectrum of self as a list.

The default algorithm uses a GAP kernel function (in C) written by Steve Linton.

INPUT:

• algorithm - None, "gap", "leon", or "binary"; defaults to "gap" except in the binary case. If "gap" then uses the GAP function, if "leon" then uses Jeffrey Leon’s software via Guava, and if "binary" then uses Sage native Cython code
• List, the spectrum

The optional algorithm ("leon") may create a stack smashing error and a traceback but should return the correct answer. It appears to run much faster than the GAP algorithm in some small examples and much slower than the GAP algorithm in other 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.spectrum()
[1, 0, 0, 7, 7, 0, 0, 1]
sage: F.<z> = GF(2^2,"z")
sage: C = codes.HammingCode(2, F); C
Linear code of length 5, dimension 3 over Finite Field in z of size 2^2
sage: C.spectrum()
[1, 0, 0, 30, 15, 18]
sage: C = codes.HammingCode(3,GF(2)); C
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: C.spectrum(algorithm="leon")   # optional - gap_packages (Guava package)
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C.spectrum(algorithm="gap")
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C.spectrum(algorithm="binary")
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C = codes.HammingCode(3,GF(3)); C
Linear code of length 13, dimension 10 over Finite Field of size 3
sage: C.spectrum() == C.spectrum(algorithm="leon")   # optional - gap_packages (Guava package)
True
sage: C = codes.HammingCode(2,GF(5)); C
Linear code of length 6, dimension 4 over Finite Field of size 5
sage: C.spectrum() == C.spectrum(algorithm="leon")   # optional - gap_packages (Guava package)
True
sage: C = codes.HammingCode(2,GF(7)); C
Linear code of length 8, dimension 6 over Finite Field of size 7
sage: C.spectrum() == C.spectrum(algorithm="leon")   # optional - gap_packages (Guava package)
True

weight_enumerator(names='xy', name2=None)

Returns the weight enumerator of the code.

INPUT:

• names - String of length 2, containing two variable names (default: "xy"). Alternatively, it can be a variable name or a string, or a tuple of variable names or strings.
• name2 - string or symbolic variable (default: None). If name2 is provided then it is assumed that names contains only one variable.

OUTPUT:

• Polynomial over $$\QQ$$

EXAMPLES:

sage: C = codes.HammingCode(3,GF(2))
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: (var1, var2) = var('var1, var2')
sage: C.weight_enumerator((var1, var2))
var1^7 + 7*var1^4*var2^3 + 7*var1^3*var2^4 + var2^7
sage: C.weight_enumerator(var1, var2)
var1^7 + 7*var1^4*var2^3 + 7*var1^3*var2^4 + var2^7

zero()

Return the zero vector.

EXAMPLES:

sage: C = codes.HammingCode(3, GF(2))
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(3,GF(2))
sage: C.zeta_function()
(2/5*T^2 + 2/5*T + 1/5)/(2*T^2 - 3*T + 1)

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(3,GF(2))
sage: C.zeta_polynomial()
2/5*T^2 + 2/5*T + 1/5
sage: C = best_known_linear_code(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(4,GF(2))
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:

• I. Duursma, “From weight enumerators to zeta functions”, in Discrete Applied Mathematics, vol. 111, no. 1-2, pp. 55-73, 2001.
sage.coding.linear_code.LinearCodeFromVectorSpace(V, d=None)

Simply converts a vector subspace $$V$$ of $$GF(q)^n$$ into a $$LinearCode$$.

INPUT:

• V – The vector space
• d – (Optional, default: None) the minimum distance of the code, if known. This is an optional parameter.

Note

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

EXAMPLES:

sage: V = VectorSpace(GF(2), 8)
sage: L = V.subspace([[1,1,1,1,0,0,0,0],[0,0,0,0,1,1,1,1]])
sage: C = LinearCodeFromVectorSpace(L)
sage: C.gen_mat()
[1 1 1 1 0 0 0 0]
[0 0 0 0 1 1 1 1]
sage: C.minimum_distance()
4


Here, we provide the minimum distance of the code.:

sage: C = LinearCodeFromVectorSpace(L, d=4)
sage: C.minimum_distance()
4

sage.coding.linear_code.best_known_linear_code(n, k, F)

Returns the best known (as of 11 May 2006) linear code of length n, dimension k over field F. The function uses the tables described in bounds_minimum_distance to construct this code.

This does not require an internet connection.

EXAMPLES:

sage: best_known_linear_code(10,5,GF(2))    # long time; optional - gap_packages (Guava package)
Linear code of length 10, dimension 5 over Finite Field of size 2
sage: gap.eval("C:=BestKnownLinearCode(10,5,GF(2))")     # long time; optional - gap_packages (Guava package)
'a linear [10,5,4]2..4 shortened code'


This means that best possible binary linear code of length 10 and dimension 5 is a code with minimum distance 4 and covering radius somewhere between 2 and 4. Use minimum_distance_why(10,5,GF(2)) or print bounds_minimum_distance(10,5,GF(2)) for further details.

sage.coding.linear_code.best_known_linear_code_www(n, k, F, verbose=False)

Explains the construction of the best known linear code over GF(q) with length n and dimension k, courtesy of the www page http://www.codetables.de/.

INPUT:

• n - Integer, the length of the code
• k - Integer, the dimension of the code
• F - Finite field, of order 2, 3, 4, 5, 7, 8, or 9
• verbose - Bool (default: False)

OUTPUT:

• Text about why the bounds are as given

EXAMPLES:

sage: L = best_known_linear_code_www(72, 36, GF(2)) # optional - internet
sage: print L                                       # optional - internet
Construction of a linear code
[72,36,15] over GF(2):
[1]:  [73, 36, 16] Cyclic Linear Code over GF(2)
CyclicCode of length 73 with generating polynomial x^37 + x^36 + x^34 +
x^33 + x^32 + x^27 + x^25 + x^24 + x^22 + x^21 + x^19 + x^18 + x^15 + x^11 +
x^10 + x^8 + x^7 + x^5 + x^3 + 1
[2]:  [72, 36, 15] Linear Code over GF(2)
Puncturing of [1] at 1


This function raises an IOError if an error occurs downloading data or parsing it. It raises a ValueError if the q input is invalid.

AUTHORS:

• Steven Sivek (2005-11-14)
• David Joyner (2008-03)
sage.coding.linear_code.bounds_minimum_distance(n, k, F)

Calculates a lower and upper bound for the minimum distance of an optimal linear code with word length n and dimension k over the field F.

The function returns a record with the two bounds and an explanation for each bound. The function Display can be used to show the explanations.

The values for the lower and upper bound are obtained from a table constructed by Cen Tjhai for GUAVA, derived from the table of Brouwer. (See http://www.win.tue.nl/ aeb/voorlincod.html or use the Sage function minimum_distance_why for the most recent data.) These tables contain lower and upper bounds for $$q=2$$ (when n <= 257), $$q=3$$ (when n <= 243), $$q=4$$ (n <= 256). (Current as of 11 May 2006.) For codes over other fields and for larger word lengths, trivial bounds are used.

This does not require an internet connection. The format of the output is a little non-intuitive. Try bounds_minimum_distance(10,5,GF(2)) for an example.

This function requires optional GAP package (Guava).

EXAMPLES:

sage: print bounds_minimum_distance(10,5,GF(2)) # optional - gap_packages (Guava package)
rec(
construction :=
[ <Operation "ShortenedCode">,
[
[ <Operation "UUVCode">,
[
[ <Operation "DualCode">,
[ [ <Operation "RepetitionCode">, [ 8, 2 ] ] ] ],
[ <Operation "UUVCode">,
[
[ <Operation "DualCode">,
[ [ <Operation "RepetitionCode">, [ 4, 2 ] ] ] ]
, [ <Operation "RepetitionCode">, [ 4, 2 ] ] ] ]
] ], [ 1, 2, 3, 4, 5, 6 ] ] ],
k := 5,
lowerBound := 4,
lowerBoundExplanation := ...
n := 10,
q := 2,
references := rec(
),
upperBound := 4,
upperBoundExplanation := ... )

sage.coding.linear_code.code2leon(C)

Writes a file in Sage’s temp directory representing the code C, returning the absolute path to the file. This is the Sage translation of the GuavaToLeon command in Guava’s codefun.gi file.

INPUT:

• C - a linear code (over GF(p), p < 11)

OUTPUT:

• Absolute path to the file written

EXAMPLES:

sage: C = codes.HammingCode(3,GF(2)); C
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: file_loc = sage.coding.linear_code.code2leon(C)
sage: f = open(file_loc); print f.read()
LIBRARY code;
code=seq(2,4,7,seq(
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
));
FINISH;
sage: f.close()

sage.coding.linear_code.min_wt_vec_gap(Gmat, n, k, F, algorithm=None)

Returns a minimum weight vector of the code generated by Gmat.

Uses C programs written by Steve Linton in the kernel of GAP, so is fairly fast. The option algorithm="guava" requires Guava. The default algorithm requires GAP but not Guava.

INPUT:

• Gmat - String representing a GAP generator matrix G of a linear code
• n - Length of the code generated by G
• k - Dimension of the code generated by G
• F - Base field

OUTPUT:

• Minimum weight vector of the code generated by Gmat

REMARKS:

• The code in the default case allows one (for free) to also compute the message vector $$m$$ such that $$m\*G = v$$, and the (minimum) distance, as a triple. however, this output is not implemented.
• The binary case can presumably be done much faster using Robert Miller’s code (see the docstring for the spectrum method). This is also not (yet) implemented.

EXAMPLES:

sage: Gstr = "Z(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: sage.coding.linear_code.min_wt_vec_gap(Gstr,7,4,GF(2))
(0, 1, 0, 1, 0, 1, 0)


This output is different but still a minimum weight vector:

sage: sage.coding.linear_code.min_wt_vec_gap(Gstr,7,4,GF(2),algorithm="guava")    # optional - gap_packages (Guava package)
(0, 0, 1, 0, 1, 1, 0)


Here Gstr is a generator matrix of the Hamming [7,4,3] binary code.

AUTHORS:

• David Joyner (11-2005)
sage.coding.linear_code.self_orthogonal_binary_codes(n, k, b=2, parent=None, BC=None, equal=False, in_test=None)

Returns a Python iterator which generates a complete set of representatives of all permutation equivalence classes of self-orthogonal binary linear codes of length in [1..n] and dimension in [1..k].

INPUT:

• n - Integer, maximal length
• k - Integer, maximal dimension
• b - Integer, requires that the generators all have weight divisible by b (if b=2, all self-orthogonal codes are generated, and if b=4, all doubly even codes are generated). Must be an even positive integer.
• parent - Used in recursion (default: None)
• BC - Used in recursion (default: None)
• equal - If True generates only [n, k] codes (default: False)
• in_test - Used in recursion (default: None)

EXAMPLES:

Generate all self-orthogonal codes of length up to 7 and dimension up to 3:

sage: for B in self_orthogonal_binary_codes(7,3):
...    print B
...
Linear code of length 2, dimension 1 over Finite Field of size 2
Linear code of length 4, dimension 2 over Finite Field of size 2
Linear code of length 6, dimension 3 over Finite Field of size 2
Linear code of length 4, dimension 1 over Finite Field of size 2
Linear code of length 6, dimension 2 over Finite Field of size 2
Linear code of length 6, dimension 2 over Finite Field of size 2
Linear code of length 7, dimension 3 over Finite Field of size 2
Linear code of length 6, dimension 1 over Finite Field of size 2


Generate all doubly-even codes of length up to 7 and dimension up to 3:

sage: for B in self_orthogonal_binary_codes(7,3,4):
...    print B; print B.gen_mat()
...
Linear code of length 4, dimension 1 over Finite Field of size 2
[1 1 1 1]
Linear code of length 6, dimension 2 over Finite Field of size 2
[1 1 1 1 0 0]
[0 1 0 1 1 1]
Linear code of length 7, dimension 3 over Finite Field of size 2
[1 0 1 1 0 1 0]
[0 1 0 1 1 1 0]
[0 0 1 0 1 1 1]


Generate all doubly-even codes of length up to 7 and dimension up to 2:

sage: for B in self_orthogonal_binary_codes(7,2,4):
...    print B; print B.gen_mat()
Linear code of length 4, dimension 1 over Finite Field of size 2
[1 1 1 1]
Linear code of length 6, dimension 2 over Finite Field of size 2
[1 1 1 1 0 0]
[0 1 0 1 1 1]


Generate all self-orthogonal codes of length equal to 8 and dimension equal to 4:

sage: for B in self_orthogonal_binary_codes(8, 4, equal=True):
...     print B; print B.gen_mat()
Linear code of length 8, dimension 4 over Finite Field of size 2
[1 0 0 1 0 0 0 0]
[0 1 0 0 1 0 0 0]
[0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 1 1]
Linear code of length 8, dimension 4 over Finite Field of size 2
[1 0 0 1 1 0 1 0]
[0 1 0 1 1 1 0 0]
[0 0 1 0 1 1 1 0]
[0 0 0 1 0 1 1 1]


Since all the codes will be self-orthogonal, b must be divisible by 2:

sage: list(self_orthogonal_binary_codes(8, 4, 1, equal=True))
Traceback (most recent call last):
...
ValueError: b (1) must be a positive even integer.

sage.coding.linear_code.wtdist_gap(Gmat, n, F)

INPUT:

• Gmat - String representing a GAP generator matrix G of a linear code
• n - Integer greater than 1, representing the number of columns of G (i.e., the length of the linear code)
• F - Finite field (in Sage), base field the code

OUTPUT:

• Spectrum of the associated code

EXAMPLES:

sage: Gstr = 'Z(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: F = GF(2)
sage: sage.coding.linear_code.wtdist_gap(Gstr, 7, F)
[1, 0, 0, 7, 7, 0, 0, 1]


Here Gstr is a generator matrix of the Hamming [7,4,3] binary code.

ALGORITHM:

Uses C programs written by Steve Linton in the kernel of GAP, so is fairly fast.

AUTHORS:

• David Joyner (2005-11)

Index of Codes

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Linear code constructions