VERSION: 1.2
Let be a finite field. Here, we will denote the finite field with
elements by
. A subspace of
(with the standard basis) is
called a linear code of length
. If its dimension is denoted
then we
typically store a basis of
as a
matrix, with rows the basis
vectors. It is called the generator matrix of
. The rows of the parity
check matrix of
are a basis for the code,
called the dual space of .
If then
is called a binary code. If
then
is
called a
-ary code. The elements of a code
are called codewords.
The symmetric group acts on
by permuting coordinates. If an
element
sends a code
of length
to itself (in other words,
every codeword of
is sent to some other codeword of
) then
is
called a permutation automorphism of
. The (permutation) automorphism
group is denoted
.
This file contains
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
To be added:
REFERENCES:
AUTHORS:
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)
sage: C == loads(dumps(C))
True
Bases: sage.modules.module.Module
A class for linear codes over a finite field or finite ring. Each instance
is a linear code determined by a generator matrix (i.e., a
matrix of (full) rank
,
over a finite field
.
INPUT:
OUTPUT:
The linear code of length over
having
as a generator matrix.
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]
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:
Assmus and Mattson Theorem (section 8.4, page 303 of [HP]): Let
be the weights of the codewords in a binary
linear
code
, and let
be
the weights of the codewords in its dual
code
.
Fix a
,
, and let
Assume .
A block design is a pair , where
is a non-empty finite set
of
elements called points, and
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
points.
design without
repeated blocks is called a simple block design. If every subset of
points of size
is contained in exactly
blocks the block
design is called a
design (or simply a
-design
when the parameters are not specified). When
then the
block design is called a
Steiner system.
In the Assmus and Mattson Theorem (1), is the set
of coordinate locations and
is the set
of supports of the codewords of
of weight
. Therefore, the
parameters of the
-design for
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 (of course
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 = 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 hamming_weight(c)==8]; len(blocks) # long time computation
759
REFERENCE:
This only applies to linear binary codes and returns its (permutation)
automorphism group. In other words, if the code has length
then it returns the subgroup of the symmetric group
:
where acts on
by permuting coordinates.
EXAMPLES:
sage: C = HammingCode(3,GF(2))
sage: G = C.automorphism_group_binary_code(); G
Permutation Group with generators [(3,4)(5,6), (3,5)(4,6), (2,3)(5,7), (1,2)(5,6)]
sage: G.order()
168
Returns the i-th binomial moment of the -code
:
where is the dimension of the shortened code
,
. (The normalized binomial moment is
.) In other words,
is isomorphic to the subcode of C of codewords supported on S.
EXAMPLES:
sage: C = HammingCode(3,GF(2))
sage: C.binomial_moment(2)
0
sage: C.binomial_moment(4) # long time
35
Warning
This is slow.
REFERENCE:
Returns the characteristic polynomial of a linear code, as defined in van Lint’s text [vL].
EXAMPLES:
sage: C = ExtendedBinaryGolayCode()
sage: C.characteristic_polynomial()
-4/3*x^3 + 64*x^2 - 2816/3*x + 4096
REFERENCES:
Returns the check matrix of self.
EXAMPLES:
sage: C = 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 1 0 1 0]
[0 1 0 1 0 1 1]
[0 0 1 1 0 0 1]
[0 0 0 0 1 1 1]
sage: C.check_mat()
[1 0 0 1 1 0 1]
[0 1 0 1 0 1 1]
[0 0 1 1 1 1 0]
sage: Cperp.check_mat()
[1 0 0 1 0 1 0]
[0 1 0 1 0 1 1]
[0 0 1 1 0 0 1]
[0 0 0 0 1 1 1]
sage: Cperp.gen_mat()
[1 0 0 1 1 0 1]
[0 1 0 1 0 1 1]
[0 0 1 1 1 1 0]
Returns the Chinen zeta polynomial of the code.
EXAMPLES:
sage: C = 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 = 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:
Wraps Guava’s CoveringRadius command.
The covering radius of a linear code is the smallest number
with the property that each element
of the ambient vector space
of
has at most a distance
to the code
. So for each
vector
there must be an element
of
with
.
A binary linear code with reasonable small covering radius is often
referred to as a covering code.
For example, if is a perfect code, the covering radius is equal
to
, the number of errors the code can correct, where
,
with
the minimum distance of
.
EXAMPLES:
sage: C = HammingCode(5,GF(2))
sage: C.covering_radius() # requires optional GAP package Guava
1
Decodes the received vector right to an element in this code.
Optional methods are “guava”, “nearest neighbor” or “syndrome”. The method="guava" wraps GUAVA’s Decodeword. Hamming codes have a special decoding algorithm; otherwise, "syndrome" decoding is used.
INPUT:
OUTPUT:
EXAMPLES:
sage: C = 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, 0, 0, 1, 1, 0, 1)
sage: C.decode(v1,method="nearest neighbor")
(1, 0, 0, 1, 1, 0, 1)
sage: C.decode(v1,method="guava") # requires optional GAP package Guava
(1, 0, 0, 1, 1, 0, 1)
sage: v2 = matrix([[a,a,F(0),a,a,F(0),a]])
sage: C.decode(v2)
(1, 0, 0, 1, 1, 0, 1)
sage: v3 = vector([a,a,F(0),a,a,F(0),a])
sage: c = C.decode(v3); c
(1, 0, 0, 1, 1, 0, 1)
sage: c in C
True
sage: C = HammingCode(2,GF(5))
sage: v = vector(GF(5),[1,0,0,2,1,0])
sage: C.decode(v)
(2, 0, 0, 2, 1, 0)
sage: F = GF(4,"a")
sage: C = HammingCode(2,F)
sage: v = vector(F, [1,0,0,a,1])
sage: C.decode(v)
(1, 0, 0, 1, 1)
sage: C.decode(v, method="nearest neighbor")
(1, 0, 0, 1, 1)
sage: C.decode(v, method="guava") # requires optional GAP package Guava
(1, 0, 0, 1, 1)
Does not work for very long codes since the syndrome table grows too large.
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
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 = 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
Returns the divisor of a code, which is the smallest integer
such that each
iff
is divisible by
.
EXAMPLES:
sage: C = ExtendedBinaryGolayCode()
sage: C.divisor() # Type II self-dual
4
sage: C = QuadraticResidueCodeEvenPair(17,GF(2))[0]
sage: C.divisor()
2
This computes the dual code of the code
,
Does not call GAP.
EXAMPLES:
sage: C = HammingCode(3,GF(2))
sage: C.dual_code()
Linear code of length 7, dimension 3 over Finite Field of size 2
sage: C = 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
If self is a linear code of length defined over
then this
returns the code of length
where the last digit
satisfies
the check condition
. If self is an
binary code then the extended code
is an
code, where
(if d is even) and
(if
is odd).
EXAMPLES:
sage: C = 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
If self is a linear code defined over and
is a subfield
with Galois group
then this returns the
-module
containing
.
EXAMPLES:
sage: C = 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()[1]
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
Return a generator matrix of this code.
EXAMPLES:
sage: C1 = HammingCode(3,GF(2))
sage: C1.gen_mat()
[1 0 0 1 0 1 0]
[0 1 0 1 0 1 1]
[0 0 1 1 0 0 1]
[0 0 0 0 1 1 1]
sage: C2 = HammingCode(2,GF(4,"a"))
sage: C2.gen_mat()
[ 1 0 0 1 1]
[ 0 1 0 1 a + 1]
[ 0 0 1 1 a]
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]
Returns the generators of this code as a list of vectors.
EXAMPLES:
sage: C = HammingCode(3,GF(2))
sage: C.gens()
[(1, 0, 0, 1, 0, 1, 0), (0, 1, 0, 1, 0, 1, 1), (0, 0, 1, 1, 0, 0, 1), (0, 0, 0, 0, 1, 1, 1)]
Returns the “Duursma genus” of the code, .
EXAMPLES:
sage: C1 = HammingCode(3,GF(2)); C1
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: C1.genus()
1
sage: C2 = 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, .
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:
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]
Checks if self is equal to its Galois closure.
EXAMPLES:
sage: C = HammingCode(3,GF(4,"a"))
sage: C.is_galois_closed()
False
Returns if
is an element of
(
= length of self) and
if
is an automorphism of self.
EXAMPLES:
sage: C = 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
Returns True if self and other are permutation equivalent codes and False otherwise.
The method="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 = CyclicCodeFromGeneratingPolynomial(7,g); C1
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: C2 = 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,method="verbose")
(True, (4,6,5,7))
sage: C1 = RandomLinearCode(10,5,GF(2))
sage: C2 = RandomLinearCode(10,5,GF(3))
sage: C1.is_permutation_equivalent(C2)
False
Returns True if the code is self-dual (in the usual Hamming inner product) and False otherwise.
EXAMPLES:
sage: C = ExtendedBinaryGolayCode()
sage: C.is_self_dual()
True
sage: C = HammingCode(3,GF(2))
sage: C.is_self_dual()
False
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 = ExtendedBinaryGolayCode()
sage: C.is_self_orthogonal()
True
sage: C = HammingCode(3,GF(2))
sage: C.is_self_orthogonal()
False
sage: C = QuasiQuadraticResidueCode(11) # requires optional GAP package Guava
sage: C.is_self_orthogonal() # requires optional GAP package Guava
True
Returns True if self is a subcode of other.
EXAMPLES:
sage: C1 = 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 = 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
Returns the length of this code.
EXAMPLES:
sage: C = HammingCode(3,GF(2))
sage: C.length()
7
Return a list of all elements of this linear code.
EXAMPLES:
sage: C = HammingCode(3,GF(2))
sage: Clist = C.list()
sage: Clist[5]; Clist[5] in C
(1, 0, 1, 0, 0, 1, 1)
True
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 method="guava" is set and
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.
INPUT:
OUTPUT:
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
sage: C.minimum_distance(method="guava") # requires optional GAP package Guava
3
sage: C = 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
This shows that trac ticket #6486 has been resolved:
sage: G = matrix(GF(2),[[0,0,0]])
sage: C = LinearCode(G)
sage: C.minimum_distance()
...
ValueError: this linear code contains no non-zero vector
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.automorphism_group_binary_code()
Now type “C.module_composition_factors(gp)” to get the record printed.
If is an
code over
, this function computes the
subgroup
of all permutation automorphisms of
.
The binary case always uses the (default) partition refinement method
of Robert Miller.
INPUT:
OUTPUT:
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
A less easy example involves showing that the permutation
automorphism group of the extended ternary Golay code is the
Mathieu group .
sage: C = ExtendedTernaryGolayCode()
sage: M11 = MathieuGroup(11)
sage: M11.order()
7920
sage: G = C.permutation_automorphism_group() # this should take < 5 seconds
sage: G.is_isomorphic(M11) # this should take < 5 seconds
True
In the binary case, uses sage.coding.binary_code:
sage: C = ExtendedBinaryGolayCode()
sage: G = C.permutation_automorphism_group()
sage: G.order()
244823040
In the non-binary case:
sage: C = HammingCode(2,GF(3)); C
Linear code of length 4, dimension 2 over Finite Field of size 3
sage: C.permutation_automorphism_group(method="partition")
Permutation Group with generators [(1,2,3)]
sage: C = HammingCode(2,GF(4,"z")); C
Linear code of length 5, dimension 3 over Finite Field in z of size 2^2
sage: C.permutation_automorphism_group(method="partition")
Permutation Group with generators [(1,2)(3,4), (1,3)(2,4)]
sage: C.permutation_automorphism_group(method="gap") # requires optional GAP package Guava
Permutation Group with generators [(1,2)(3,4), (1,3)(2,4)]
sage: C = TernaryGolayCode()
sage: C.permutation_automorphism_group(method="gap") # requires optional GAP package Guava
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 method="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(method="gap").
Returns the permuted code, which is equivalent to self via the column permutation p.
EXAMPLES:
sage: C = HammingCode(3,GF(2))
sage: G = C.automorphism_group_binary_code(); G
Permutation Group with generators [(3,4)(5,6), (3,5)(4,6), (2,3)(5,7), (1,2)(5,6)]
sage: g = G("(2,3)(5,7)")
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
Returns the code punctured at the positions ,
. If this code
is of length
in
GF(q) then the code
obtained from
by puncturing at the
positions in
is the code of length
consisting of codewords
of
which have their
coordinate deleted if
and
left alone if
:
where . In particular, if
then
is simply the code obtainen from
by deleting the
coordinate of each codeword. The code
is called the punctured
code at
. The dimension of
can decrease if
.
INPUT:
OUTPUT:
EXAMPLES:
sage: C = HammingCode(3,GF(2))
sage: C.punctured([1,2])
Linear code of length 5, dimension 4 over Finite Field of size 2
Returns a random codeword.
OUTPUT:
EXAMPLES:
sage: C = HammingCode(3,GF(4,'a'))
sage: Cc = C.galois_closure(GF(2))
sage: c = C.gen_mat()[1]
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
If C is a linear [n,k,d] code then this function returns a
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:
EXAMPLES:
sage: C = HammingCode(3,GF(2))
sage: C.gen_mat()
[1 0 0 1 0 1 0]
[0 1 0 1 0 1 1]
[0 0 1 1 0 0 1]
[0 0 0 0 1 1 1]
sage: C.redundancy_matrix()
[1 1 0]
[1 1 1]
[1 0 1]
[0 1 1]
sage: C.standard_form()[0].gen_mat()
[1 0 0 0 1 1 0]
[0 1 0 0 1 1 1]
[0 0 1 0 1 0 1]
[0 0 0 1 0 1 1]
sage: C = HammingCode(2,GF(3))
sage: C.gen_mat()
[1 0 2 2]
[0 1 2 1]
sage: C.redundancy_matrix()
[2 2]
[2 1]
Returns the Duursma data and
of this formally s.d. code
and the type number
in (1,2,3,4). Does not check if this code
is actually sd.
INPUT:
OUTPUT:
REFERENCES:
[D] | (1, 2, 3) I. Duursma, “Extremal weight enumerators and ultraspherical polynomials” |
INPUT:
OUTPUT:
REFERENCES:
Returns the Duursma zeta function of a self-dual code using the construction in [D].
INPUT:
OUTPUT:
EXAMPLES:
sage: C1 = 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 .
REFERENCES:
Returns the code shortened at the positions L, where
.
Consider the subcode consisting of all codewords
which
satisfy
for all
. The punctured code
is
called the shortened code on
and is denoted
. The code
constructed is actually only isomorphic to the shortened code defined
in this way.
By Theorem 1.5.7 in [HP], is
. This is used
in the construction below.
INPUT:
OUTPUT:
EXAMPLES:
sage: C = HammingCode(3,GF(2))
sage: C.shortened([1,2])
Linear code of length 5, dimension 2 over Finite Field of size 2
Returns the spectrum of self as a list.
The default method uses a GAP kernel function (in C) written by Steve Linton.
INPUT:
The optional method ("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 method in some small examples and much slower than the GAP method 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 = 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 = HammingCode(3,GF(2)); C
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: C.spectrum(method="leon") # requires optional GAP package Guava
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C.spectrum(method="gap")
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C.spectrum(method="binary")
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C = HammingCode(3,GF(3)); C
Linear code of length 13, dimension 10 over Finite Field of size 3
sage: C.spectrum() == C.spectrum(method="leon") # requires optional GAP package Guava
True
sage: C = HammingCode(2,GF(5)); C
Linear code of length 6, dimension 4 over Finite Field of size 5
sage: C.spectrum() == C.spectrum(method="leon") # requires optional GAP package Guava
True
sage: C = HammingCode(2,GF(7)); C
Linear code of length 8, dimension 6 over Finite Field of size 7
sage: C.spectrum() == C.spectrum(method="leon") # requires optional GAP package Guava
True
Returns the standard form of this linear code.
An linear code with generator matrix
in standard form is
the row-reduced echelon form of
is
, where
denotes the
identity matrix and
is a
block. This
method returns a pair
where
is a code permutation
equivalent to self and
in
, with
the length of
,
is the permutation sending self to
. This does not call GAP.
Thanks to Frank Luebeck for (the GAP version of) this code.
EXAMPLES:
sage: C = HammingCode(3,GF(2))
sage: C.gen_mat()
[1 0 0 1 0 1 0]
[0 1 0 1 0 1 1]
[0 0 1 1 0 0 1]
[0 0 0 0 1 1 1]
sage: Cs,p = C.standard_form()
sage: p
(4,5)
sage: Cs
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: Cs.gen_mat()
[1 0 0 0 1 1 0]
[0 1 0 0 1 1 1]
[0 0 1 0 1 0 1]
[0 0 0 1 0 1 1]
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: G; C.standard_form()[0].gen_mat()
[1 0 0 0 1 1 0]
[0 1 0 1 0 1 0]
[0 0 0 0 0 0 1]
[1 0 0 0 1 1 0]
[0 1 0 1 0 1 0]
[0 0 1 0 0 0 0]
sage: C.standard_form()[1]
(3,7)
Returns the set of indices where
is nonzero, where
spectrum(self) =
.
OUTPUT:
EXAMPLES:
sage: C = HammingCode(3,GF(2))
sage: C.spectrum()
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C.support()
[0, 3, 4, 7]
Returns the spectrum of self as a list.
The default method uses a GAP kernel function (in C) written by Steve Linton.
INPUT:
The optional method ("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 method in some small examples and much slower than the GAP method 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 = 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 = HammingCode(3,GF(2)); C
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: C.spectrum(method="leon") # requires optional GAP package Guava
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C.spectrum(method="gap")
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C.spectrum(method="binary")
[1, 0, 0, 7, 7, 0, 0, 1]
sage: C = HammingCode(3,GF(3)); C
Linear code of length 13, dimension 10 over Finite Field of size 3
sage: C.spectrum() == C.spectrum(method="leon") # requires optional GAP package Guava
True
sage: C = HammingCode(2,GF(5)); C
Linear code of length 6, dimension 4 over Finite Field of size 5
sage: C.spectrum() == C.spectrum(method="leon") # requires optional GAP package Guava
True
sage: C = HammingCode(2,GF(7)); C
Linear code of length 8, dimension 6 over Finite Field of size 7
sage: C.spectrum() == C.spectrum(method="leon") # requires optional GAP package Guava
True
Returns the weight enumerator of the code.
INPUT:
OUTPUT:
EXAMPLES:
sage: C = 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
Returns the Duursma zeta function of the code.
INPUT:
OUTPUT:
EXAMPLES:
sage: C = HammingCode(3,GF(2))
sage: C.zeta_function()
(2/5*T^2 + 2/5*T + 1/5)/(2*T^2 - 3*T + 1)
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:
OUTPUT:
EXAMPLES:
sage: C = 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)) # requires optional GAP package Guava
sage: C.minimum_distance() # requires optional GAP package Guava
3
sage: C.zeta_polynomial() # requires optional GAP package Guava
2/5*T^2 + 2/5*T + 1/5
sage: C = 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:
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 and requires optional GAP package Guava
Linear code of length 10, dimension 5 over Finite Field of size 2
sage: gap.eval("C:=BestKnownLinearCode(10,5,GF(2))") # long time and requires optional GAP package Guava
'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.
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:
OUTPUT:
EXAMPLES:
sage: L = best_known_linear_code_www(72, 36, GF(2)) # requires internet, optional
sage: print L # requires internet, optional
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
last modified: 2002-03-20
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:
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 (when n <= 257),
(when n <= 243),
(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).
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:
OUTPUT:
EXAMPLES:
sage: C = 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(
1,0,0,1,0,1,0,
0,1,0,1,0,1,1,
0,0,1,1,0,0,1,
0,0,0,0,1,1,1
));
FINISH;
sage: f.close()
Returns the Hamming weight of the vector v, which is the number of non-zero entries.
INPUT:
OUTPUT:
EXAMPLES:
sage: hamming_weight(vector(GF(2),[0,0,1]))
1
sage: hamming_weight(vector(GF(2),[0,0,0]))
0
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 method="guava" requires Guava. The default method requires GAP but not Guava.
INPUT:
OUTPUT:
REMARKS:
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),method="guava") # requires optional GAP package Guava
(0, 0, 1, 0, 1, 1, 0)
Here Gstr is a generator matrix of the Hamming [7,4,3] binary code.
AUTHORS:
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:
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))
...
ValueError: b (1) must be a positive even integer.
INPUT:
OUTPUT:
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: