Module: sage.coding.code_constructions
Author Log:
This file contains contructions of error-correcting codes which are pure Python/SAGE and not obtained from wrapping GUAVA functions. The GUAVA wrappers are in guava.py.
Let
be a finite field with
elements. Here's a constructive
definition of a cyclic code of length
.
Let
be a positive integer relatively prime to
and
let
be a primitive
-th root of unity. Each generator
polynomial
of a cyclic code
of length
has a factorization
of the form
where
* BCHCode - A 'Bose-Chaudhuri-Hockenghem code' (or BCH code for short) is the
largest possible cyclic code of length n over field F=GF(q), whose generator
polynomial has zeros (which contain the set)
,
where a is a primitive
root of unity in the splitting field
,
b is an integer
and m is the multiplicative order of q modulo n.
* BinaryGolayCode, ExtendedBinaryGolayCode, TernaryGolayCode, ExtendedTernaryGolayCode the well-known "extremal" Golay codes, http://en.wikipedia.org/wiki/Golay_code
* cyclic codes - CyclicCodeFromGeneratingPolynomial (= CyclicCode), CyclicCodeFromCheckPolynomial, http://en.wikipedia.org/wiki/Cyclic_code
* DuadicCodeEvenPair, DuadicCodeOddPair: Constructs the "even (resp. odd) pair" of duadic codes associated to the "splitting" S1, S2 of n. This is a special type of cyclic code whose generator is determined by S1, S2. See chapter 6 in [HP].
* HammingCode - the well-known Hamming code, http://en.wikipedia.org/wiki/Hamming_code
* LinearCodeFromCheckMatrix - for specifing the code using the check matrix instead of the generator matrix.
* QuadraticResidueCodeEvenPair, QuadraticResidueCodeOddPair: Quadratic residue codes of a given odd prime length and base ring either don't exist at all or occur as 4-tuples - a pair of ``odd-like'' codes and a pair of ``even-like'' codes. If n > 2 is prime then (Theorem 6.6.2 in [HP]) a QR code exists over GF(q) iff q is a quadratic residue mod n. Here they are constructed as "even-like" duadic codes associated the splitting (Q,N) mod n, where Q is the set of non-zero quadratic residues and N is the non-residues. QuadraticResidueCode (a special case) and ExtendedQuadraticResidueCode are included as well.
* RandomLinearCode - Repeatedly applies SAGE's random_element applied to the ambient MatrixSpace of the generator matrix until a full rank matrix is found.
* ReedSolomonCode - Given a finite field
of order
, let
and
be chosen such that
. Pick
distinct elements of
, denoted
.
Then, the codewords are obtained by evaluating every polynomial
in
of degree less than
at each
.
* ToricCode - Let
denote a list of lattice points in
and let
denote a listing of all points in
. Put
and let
denote the dimension of the vector space of functions
. The associated toric code
is the
evaluation code which is the image of the evaluation map
, where
is the multi-index notation.
* WalshCode - a binary linear
code related to Hadamard matrices.
http://en.wikipedia.org/wiki/Walsh_code
Please see the docstrings below for further details.
Module-level Functions
n, delta, F, [b=0]) |
A 'Bose-Chaudhuri-Hockenghem code' (or BCH code for short) is the
largest possible cyclic code of length n over field F=GF(q), whose generator
polynomial has zeros (which contain the set)
,
where a is a primitive
root of unity in the splitting field
,
b is an integer
and m is the multiplicative order of q modulo n.
(The integers
typically lie in the range
.) The integer
is called the "designed distance". The length n of the code and the
size q of the base field must be relatively prime. The generator polynomial is equal
to the least common multiple of the minimal polynomials of the elements of the set
above.
Special cases are b=1 (resulting codes are called 'narrow-sense' BCH codes), and
(known as 'primitive' BCH codes).
It may happen that several values of delta give rise to the same BCH code.
Thlargest one is called the Bose distance of the code. The true minimum distance, d, of
the code is greater than or equal to the Bose distance, so
.
sage: FF.<a> = GF(3^2,"a") sage: x = PolynomialRing(FF,"x").gen() sage: L = [b.minpoly() for b in [a,a^2,a^3]]; g = LCM(L) sage: f = x^(8)-1 sage: g.divides(f) True sage: C = CyclicCode(8,g); C Linear code of length 8, dimension 4 over Finite Field of size 3 sage: C.minimum_distance() 4 sage: C = BCHCode(8,3,GF(3),1); C Linear code of length 8, dimension 4 over Finite Field of size 3 sage: C.minimum_distance() 4 sage: C = BCHCode(8,3,GF(3)); C Linear code of length 8, dimension 3 over Finite Field of size 3 sage: C.minimum_distance() 5
REFERENCES: [HP] W. C. Huffman, V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, 2003.
) |
BinaryGolayCode() returns a binary Golay code. This is a perfect [23,12,7] code.
It is also (equivalent to) a cyclic code, with generator polynomial
.
Extending it yields the extended Golay code (see ExtendedBinaryGolayCode).
sage: C = BinaryGolayCode() sage: C Linear code of length 23, dimension 12 over Finite Field of size 2 sage: C.minimum_distance() # long time 7
Author: David Joyner (2007-05)
n, g, [ignore=True]) |
If g is a polynomial over GF(q) which divides
then this
constructs the code "generated by g" (ie, the code associated with
the principle ideal
in the ring
in the usual way).
The option "ignore" says to ignore the condition that
(a) the characteristic of the base field does not divide the length
(the usual assumtion in the theory of cyclic codes), and
(b)
must divide
. If ignore=True, instead of returning an error, a
code generated by
is created.
sage: P.<x> = PolynomialRing(GF(3),"x") sage: g = x-1 sage: C = CyclicCodeFromGeneratingPolynomial(4,g); C Linear code of length 4, dimension 3 over Finite Field of size 3 sage: P.<x> = PolynomialRing(GF(4,"a"),"x") sage: g = x^3+1 sage: C = CyclicCodeFromGeneratingPolynomial(9,g); C Linear code of length 9, dimension 6 over Finite Field in a of size 2^2 sage: P.<x> = PolynomialRing(GF(2),"x") sage: g = x^3+x+1 sage: C = CyclicCodeFromGeneratingPolynomial(7,g); C Linear code of length 7, dimension 4 over Finite Field of size 2 sage: C.gen_mat() [1 1 0 1 0 0 0] [0 1 1 0 1 0 0] [0 0 1 1 0 1 0] [0 0 0 1 1 0 1] sage: g = x+1 sage: C = CyclicCodeFromGeneratingPolynomial(4,g); C Linear code of length 4, dimension 3 over Finite Field of size 2 sage: C.gen_mat() [1 1 0 0] [0 1 1 0] [0 0 1 1]
On the other hand, CyclicCodeFromPolynomial(4,x) will produce
a ValueError including a traceback error message: "
must divide
".
You will also get a ValueError if you type
sage: P.<x> = PolynomialRing(GF(4,"a"),"x") sage: g = x^2+1
followed by CyclicCodeFromGeneratingPolynomial(6,g). You will also get a ValueError if you type
sage: P.<x> = PolynomialRing(GF(3),"x") sage: g = x^2-1 sage: C = CyclicCodeFromGeneratingPolynomial(5,g); C Linear code of length 5, dimension 4 over Finite Field of size 3
followed by C = CyclicCodeFromGeneratingPolynomial(5,g,False), with
a traceback message including "
must divide
".
n, h, [ignore=True]) |
If h is a polynomial over GF(q) which divides
then this
constructs the code "generated by
" (ie, the code associated with
the principle ideal
in the ring
in the usual way).
The option "ignore" says to ignore the condition that the
characteristic of the base field does nto divide the length
(the usual assumtion in the theory of cyclic codes).
sage: P.<x> = PolynomialRing(GF(3),"x") sage: C = CyclicCodeFromCheckPolynomial(4,x + 1); C Linear code of length 4, dimension 1 over Finite Field of size 3 sage: C = CyclicCodeFromCheckPolynomial(4,x^3 + x^2 + x + 1); C Linear code of length 4, dimension 3 over Finite Field of size 3 sage: C.gen_mat() [2 1 0 0] [0 2 1 0] [0 0 2 1]
n, g, [ignore=True]) |
If g is a polynomial over GF(q) which divides
then this
constructs the code "generated by g" (ie, the code associated with
the principle ideal
in the ring
in the usual way).
The option "ignore" says to ignore the condition that
(a) the characteristic of the base field does not divide the length
(the usual assumtion in the theory of cyclic codes), and
(b)
must divide
. If ignore=True, instead of returning an error, a
code generated by
is created.
sage: P.<x> = PolynomialRing(GF(3),"x") sage: g = x-1 sage: C = CyclicCodeFromGeneratingPolynomial(4,g); C Linear code of length 4, dimension 3 over Finite Field of size 3 sage: P.<x> = PolynomialRing(GF(4,"a"),"x") sage: g = x^3+1 sage: C = CyclicCodeFromGeneratingPolynomial(9,g); C Linear code of length 9, dimension 6 over Finite Field in a of size 2^2 sage: P.<x> = PolynomialRing(GF(2),"x") sage: g = x^3+x+1 sage: C = CyclicCodeFromGeneratingPolynomial(7,g); C Linear code of length 7, dimension 4 over Finite Field of size 2 sage: C.gen_mat() [1 1 0 1 0 0 0] [0 1 1 0 1 0 0] [0 0 1 1 0 1 0] [0 0 0 1 1 0 1] sage: g = x+1 sage: C = CyclicCodeFromGeneratingPolynomial(4,g); C Linear code of length 4, dimension 3 over Finite Field of size 2 sage: C.gen_mat() [1 1 0 0] [0 1 1 0] [0 0 1 1]
On the other hand, CyclicCodeFromPolynomial(4,x) will produce
a ValueError including a traceback error message: "
must divide
".
You will also get a ValueError if you type
sage: P.<x> = PolynomialRing(GF(4,"a"),"x") sage: g = x^2+1
followed by CyclicCodeFromGeneratingPolynomial(6,g). You will also get a ValueError if you type
sage: P.<x> = PolynomialRing(GF(3),"x") sage: g = x^2-1 sage: C = CyclicCodeFromGeneratingPolynomial(5,g); C Linear code of length 5, dimension 4 over Finite Field of size 3
followed by C = CyclicCodeFromGeneratingPolynomial(5,g,False), with
a traceback message including "
must divide
".
F, S1, S2) |
Constructs the "even pair" of duadic codes associated to
the "splitting" (see the docstring for is_a_splitting
for the
definition) S1, S2 of n.
WARNING?: Maybe the splitting should be associated to a sum of q-cyclotomic cosets mod n, where q is a *prime*.
sage: from sage.coding.code_constructions import is_a_splitting sage: n = 11; q = 3 sage: C = cyclotomic_cosets(q,n); C [[0], [1, 3, 4, 5, 9], [2, 6, 7, 8, 10]] sage: S1 = C[1] sage: S2 = C[2] sage: is_a_splitting(S1,S2,11) (True, 2) sage: DuadicCodeEvenPair(GF(q),S1,S2) (Linear code of length 11, dimension 5 over Finite Field of size 3, Linear code of length 11, dimension 5 over Finite Field of size 3)
F, S1, S2) |
Constructs the "odd pair" of duadic codes associated to the "splitting" S1, S2 of n.
WARNING?: Maybe the splitting should be associated to a sum of q-cyclotomic cosets mod n, where q is a *prime*.
sage: from sage.coding.code_constructions import is_a_splitting sage: n = 11; q = 3 sage: C = cyclotomic_cosets(q,n); C [[0], [1, 3, 4, 5, 9], [2, 6, 7, 8, 10]] sage: S1 = C[1] sage: S2 = C[2] sage: is_a_splitting(S1,S2,11) (True, 2) sage: DuadicCodeOddPair(GF(q),S1,S2) (Linear code of length 11, dimension 6 over Finite Field of size 3, Linear code of length 11, dimension 6 over Finite Field of size 3)
This is consistent with Theorem 6.1.3 in [HP].
) |
ExtendedBinaryGolayCode() returns the extended binary Golay code. This is a perfect [24,12,8] code. This code is self-dual.
sage: C = ExtendedBinaryGolayCode() sage: C Linear code of length 24, dimension 12 over Finite Field of size 2 sage: C.minimum_distance() 8
Author: David Joyner (2007-05)
n, F) |
The extended quadratic residue code (or XQR code) is obtained from a QR code by adding a check bit to the last coordinate. (These codes have very remarkable properties such as large automorphism groups and duality properties - see [HP], §6.6.3-6.6.4.)
Input:
sage: C1 = QuadraticResidueCode(7,GF(2)) sage: C2 = C1.extended_code() sage: C3 = ExtendedQuadraticResidueCode(7,GF(2)); C3 Linear code of length 8, dimension 4 over Finite Field of size 2 sage: C2 == C3 True sage: C = ExtendedQuadraticResidueCode(17,GF(2)) sage: C Linear code of length 18, dimension 9 over Finite Field of size 2 sage: C3 = QuadraticResidueCodeOddPair(7,GF(2))[0] sage: C3x = C3.extended_code() sage: C4 = ExtendedQuadraticResidueCode(7,GF(2)) sage: C3x == C4 True
Author: David Joyner (07-2006)
) |
ExtendedTernaryGolayCode returns a ternary Golay code. This is a self-dual perfect [12,6,6] code.
sage: C = ExtendedTernaryGolayCode() sage: C Linear code of length 12, dimension 6 over Finite Field of size 3 sage: C.minimum_distance() 6
Author: David Joyner (11-2005)
r, F) |
Implements the Hamming codes.
The
Hamming code over
is an
code
with length
, dimension
and
minimum distance
.
The parity check matrix of a Hamming code has rows consisting of
all nonzero vectors of length r in its columns, modulo a scalar factor
so no parallel columns arise. A Hamming code is a single error-correcting
code.
Input:
sage: HammingCode(3,GF(2)) Linear code of length 7, dimension 4 over Finite Field of size 2 sage: C = HammingCode(3,GF(3)); C Linear code of length 13, dimension 10 over Finite Field of size 3 sage: C.minimum_distance() 3 sage: C = HammingCode(3,GF(4,'a')); C Linear code of length 21, dimension 18 over Finite Field in a of size 2^2
H) |
A linear [n,k]-code C is uniquely determined by its generator matrix G and check matrix H. We have the following short exact sequence
![]() |
(39.1) |
sage: C = HammingCode(3,GF(2)) sage: H = C.check_mat(); H [1 0 0 1 1 0 1] [0 1 0 1 0 1 1] [0 0 1 1 1 1 0] sage: LinearCodeFromCheckMatrix(H) == C True sage: C = HammingCode(2,GF(3)) sage: H = C.check_mat(); H [1 0 2 2] [0 1 2 1] sage: LinearCodeFromCheckMatrix(H) == C True sage: C = RandomLinearCode(10,5,GF(4,"a")) sage: H = C.check_mat() sage: LinearCodeFromCheckMatrix(H) == C True
n, F) |
A quadratic residue code (or QR code) is a cyclic code whose
generator polynomial is the product of the polynomials
(
is a primitive
root of unity;
ranges over
the set of quadratic residues modulo
).
See QuadraticResidueCodeEvenPair and QuadraticResidueCodeOddPair for a more general construction.
Input:
sage: C = QuadraticResidueCode(7,GF(2)) sage: C Linear code of length 7, dimension 4 over Finite Field of size 2 sage: C = QuadraticResidueCode(17,GF(2)) sage: C Linear code of length 17, dimension 9 over Finite Field of size 2 sage: C1 = QuadraticResidueCodeOddPair(7,GF(2))[0] sage: C2 = QuadraticResidueCode(7,GF(2)) sage: C1 == C2 True sage: C1 = QuadraticResidueCodeOddPair(17,GF(2))[0] sage: C2 = QuadraticResidueCode(17,GF(2)) sage: C1 == C2 True
Author: David Joyner (11-2005)
n, F) |
Quadratic residue codes of a given odd prime length and base ring either don't exist at all or occur as 4-tuples - a pair of ``odd-like'' codes and a pair of ``even-like'' codes. If n > 2 is prime then (Theorem 6.6.2 in [HP]) a QR code exists over GF(q) iff q is a quadratic residue mod n.
They are constructed as "even-like" duadic codes associated the splitting (Q,N) mod n, where Q is the set of non-zero quadratic residues and N is the non-residues.
sage: QuadraticResidueCodeEvenPair(17,GF(13)) (Linear code of length 17, dimension 8 over Finite Field of size 13, Linear code of length 17, dimension 8 over Finite Field of size 13) sage: QuadraticResidueCodeEvenPair(17,GF(2)) (Linear code of length 17, dimension 8 over Finite Field of size 2, Linear code of length 17, dimension 8 over Finite Field of size 2) sage: QuadraticResidueCodeEvenPair(13,GF(9,"z")) (Linear code of length 13, dimension 6 over Finite Field in z of size 3^2, Linear code of length 13, dimension 6 over Finite Field in z of size 3^2) sage: C1 = QuadraticResidueCodeEvenPair(7,GF(2))[0] sage: C1.is_self_orthogonal() True sage: C2 = QuadraticResidueCodeEvenPair(7,GF(2))[1] sage: C2.is_self_orthogonal() True sage: C3 = QuadraticResidueCodeOddPair(17,GF(2))[0] sage: C4 = QuadraticResidueCodeEvenPair(17,GF(2))[1] sage: C3 == C4.dual_code() True
This is consistent with Theorem 6.6.9 and Exercise 365 in [HP].
n, F) |
Quadratic residue codes of a given odd prime length and base ring either don't exist at all or occur as 4-tuples - a pair of ``odd-like'' codes and a pair of ``even-like'' codes. If n > 2 is prime then (Theorem 6.6.2 in [HP]) a QR code exists over GF(q) iff q is a quadratic residue mod n.
They are constructed as "odd-like" duadic codes associated the splitting (Q,N) mod n, where Q is the set of non-zero quadratic residues and N is the non-residues.
sage: QuadraticResidueCodeOddPair(17,GF(13)) (Linear code of length 17, dimension 9 over Finite Field of size 13, Linear code of length 17, dimension 9 over Finite Field of size 13) sage: QuadraticResidueCodeOddPair(17,GF(2)) (Linear code of length 17, dimension 9 over Finite Field of size 2, Linear code of length 17, dimension 9 over Finite Field of size 2) sage: QuadraticResidueCodeOddPair(13,GF(9,"z")) (Linear code of length 13, dimension 7 over Finite Field in z of size 3^2, Linear code of length 13, dimension 7 over Finite Field in z of size 3^2) sage: C1 = QuadraticResidueCodeOddPair(17,GF(2))[1] sage: C1x = C1.extended_code() sage: C2 = QuadraticResidueCodeOddPair(17,GF(2))[0] sage: C2x = C2.extended_code() sage: C2x.spectrum(); C1x.spectrum() [1, 0, 0, 0, 0, 0, 102, 0, 153, 0, 153, 0, 102, 0, 0, 0, 0, 0, 1] [1, 0, 0, 0, 0, 0, 102, 0, 153, 0, 153, 0, 102, 0, 0, 0, 0, 0, 1] sage: C2x == C1x.dual_code() True sage: C3 = QuadraticResidueCodeOddPair(7,GF(2))[0] sage: C3x = C3.extended_code() sage: C3x.spectrum() [1, 0, 0, 0, 14, 0, 0, 0, 1] sage: C3x.is_self_dual() True
This is consistent with Theorem 6.6.14 in [HP].
n, k, F) |
The method used is to first construct a
matrix using SAGE's
random_element method for the MatrixSpace class. The construction is
probabilistic but should only fail extremely rarely.
Input: Integers n,k, with n>k>1, and a finite field F
Output: Returns a "random" linear code with length n, dimension k over field F.
sage: C = RandomLinearCode(30,15,GF(2)) sage: C Linear code of length 30, dimension 15 over Finite Field of size 2 sage: C = RandomLinearCode(10,5,GF(4,'a')) sage: C Linear code of length 10, dimension 5 over Finite Field in a of size 2^2
Author: David Joyner (2007-05)
n, k, F, [pts=None]) |
Given a finite field
of order
, let
and
be chosen such that
. Pick
distinct elements of
, denoted
. Then, the codewords are obtained by evaluating
every polynomial in
of degree less than
at each
:
Input: n : the length k : the dimension F : the base ring pts : (optional) list of n points in F (if None then SAGE picks n of them in the order given to the elements of F)
sage: C = ReedSolomonCode(6,4,GF(7)); C Linear code of length 6, dimension 4 over Finite Field of size 7 sage: C.minimum_distance() 3 sage: C = ReedSolomonCode(6,4,GF(8,"a")); C Linear code of length 6, dimension 4 over Finite Field in a of size 2^3 sage: C.minimum_distance() 3 sage: F.<a> = GF(3^2,"a") sage: pts = [0,1,a,a^2,2*a,2*a+1] sage: len(Set(pts)) == 6 # to make sure there are no duplicates True sage: C = ReedSolomonCode(6,4,F,pts); C Linear code of length 6, dimension 4 over Finite Field in a of size 3^2 sage: C.minimum_distance() 3
REFERENCES: [HP] W. C. Huffman, V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, 2003. [W] http://en.wikipedia.org/wiki/Reed-Solomon
) |
TernaryGolayCode returns a ternary Golay code. This is a perfect
[11,6,5] code. It is also equivalenet to a cyclic code, with
generator polynomial
.
sage: C = TernaryGolayCode() sage: C Linear code of length 11, dimension 6 over Finite Field of size 3 sage: C.minimum_distance() 5
Author: David Joyner (2007-5)
P, F) |
Let
denote a list of lattice points in
and let
denote the
set of all points in
(ordered in some fixed way). Put
and let
denote the dimension of the vector space of functions
. The associated toric code
is the
evaluation code which is the image of the evaluation map
where
Input:
sage: C = ToricCode([[0,0],[1,0],[2,0],[0,1],[1,1]],GF(7)) sage: C Linear code of length 36, dimension 5 over Finite Field of size 7 sage: C.minimum_distance() 24 sage: C = ToricCode([[-2,-2],[-1,-2],[-1,-1],[-1,0],[0,-1],[0,0],[0,1],[1,-1],[1,0]],GF(5)) sage: C Linear code of length 16, dimension 9 over Finite Field of size 5 sage: C.minimum_distance() 6 sage: C = ToricCode([ [0,0],[1,1],[1,2],[1,3],[1,4],[2,1],[2,2],[2,3],[3,1],[3,2],[4,1]],GF(8,"a")) sage: C Linear code of length 49, dimension 11 over Finite Field in a of size 2^3
This is in fact a [49,11,28] code over GF(8). If you type next
C.minimum_distance()
and wait overnight (!), you should get 28.
Author: David Joyner (07-2006)
REFERENCES: [J] D. Joyner, Toric codes over finite fields, Applicable Algebra in Engineering, Communication and Computing, 15, (2004), p. 63-79
F, n) |
m) |
Returns the binary Walsh code of length
. The matrix of codewords correspond to
a Hadamard matrix. This is a (constant rate) binary linear
code.
sage: C = WalshCode(4); C Linear code of length 16, dimension 4 over Finite Field of size 2 sage: C.minimum_distance() 8
REFERENCES: http://en.wikipedia.org/wiki/Hadamard_matrix http://en.wikipedia.org/wiki/Walsh_code
q, n, [t=None]) |
Input: q,n,t positive integers (or t=none) Some type-checking of inputs is performed.
Output: q-cyclotomic cosets mod n (or, if t<>none, the q-cyclotomic coset mod n containing t)
Let q, n be relatively print positive integers and let
. The group A acts on ZZ/nZZ by multiplication.
The orbits of this action are "cyclotomic cosets",
or more precisely "q-cyclotomic cosets mod n". Sometimes the
smallest element of the coset is called the "coset leader".
The algorithm will always return the cosets as sorted lists of
lists, so the coset leader will always be the first element
in the list.
These cosets arise in the theory of duadic codes and
minimal polynomials of finite fields. Fix a primitive
element
of
. The minimal polynomial of
over
is given by
where
sage: cyclotomic_cosets(2,11) [[0], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]] sage: cyclotomic_cosets(2,15) [[0], [1, 2, 4, 8], [3, 6, 9, 12], [5, 10], [7, 11, 13, 14]] sage: cyclotomic_cosets(2,15,5) [5, 10] sage: cyclotomic_cosets(3,16) [[0], [1, 3, 9, 11], [2, 6], [4, 12], [5, 7, 13, 15], [8], [10, 14]] sage: F.<z> = GF(2^4, "z") sage: P.<x> = PolynomialRing(F,"x") sage: a = z^5 sage: a.minimal_polynomial() x^2 + x + 1 sage: prod([x-z^i for i in [5, 10]]) x^2 + x + 1 sage: cyclotomic_cosets(3,2,0) [0] sage: cyclotomic_cosets(3,2,1) [1] sage: cyclotomic_cosets(3,2,2) [0]
This last output loks strange but is correct, since the elements of the cosets are in ZZ/nZZ and 2 = 0 in ZZ/2ZZ.
S1, S2, n) |
Input: s1, s2 are disjoint sublists partitioning [1, 2, ..., n-1] n>1 is an integer
Output: a, b where a is True or False, depending on whether S1, S2 form a "splitting" of n (ie, if there is a b>1 such that b*S1=S2 (point-wise multiplication mod n), and b is a splitting (if a = True) or 0 (if a = False)
Splittings are useful for computing idempotents in the
quotient ring
. For
sage: from sage.coding.code_constructions import is_a_splitting sage: n = 11; q = 3 sage: C = cyclotomic_cosets(q,n); C [[0], [1, 3, 4, 5, 9], [2, 6, 7, 8, 10]] sage: S1 = C[1] sage: S2 = C[2] sage: is_a_splitting(S1,S2,11) (True, 2) sage: F = GF(q) sage: P.<x> = PolynomialRing(F,"x") sage: I = Ideal(P,[x^n-1]) sage: Q.<x> = QuotientRing(P,I) sage: i1 = -sum([x^i for i in S1]); i1 2*x^9 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x sage: i2 = -sum([x^i for i in S2]); i2 2*x^10 + 2*x^8 + 2*x^7 + 2*x^6 + 2*x^2 sage: i1^2 == i1 True sage: i2^2 == i2 True sage: (1-i1)^2 == 1-i1 True sage: (1-i2)^2 == 1-i2 True
We return to dealing with polynomials (rather than elements of quotient rings), so we can construct cyclic codes:
sage: P.<x> = PolynomialRing(F,"x") sage: i1 = -sum([x^i for i in S1]) sage: i2 = -sum([x^i for i in S2]) sage: i1_sqrd = (i1^2).quo_rem(x^n-1)[1] sage: i1_sqrd == i1 True sage: i2_sqrd = (i2^2).quo_rem(x^n-1)[1] sage: i2_sqrd == i2 True sage: C1 = CyclicCodeFromGeneratingPolynomial(n,i1) sage: C2 = CyclicCodeFromGeneratingPolynomial(n,1-i2) sage: C1.dual_code() == C2 True
This is a special case of Theorem 6.4.3 in [HP].
a) |
Input: a is an element of a finite field gf(q)
Output: the element b of the smallest subfield f of gf(q) for which F(b)=a.
sage: from sage.coding.code_constructions import lift2smallest_field sage: FF.<z> = GF(3^4,"z") sage: a = z^10 sage: lift2smallest_field(a) (2*z + 1, Finite Field in z of size 3^2) sage: a = z^40 sage: lift2smallest_field(a) (2, Finite Field of size 3)
Author: John Cremona
a) |
Input: a is an element of a finite field gf(q)
Output: the element b of the smallest subfield f of gf(q) for which F(b)=a.
sage: from sage.coding.code_constructions import lift2smallest_field2 sage: FF.<z> = GF(3^4,"z") sage: a = z^40 sage: lift2smallest_field2(a) (2, Finite Field of size 3) sage: FF.<z> = GF(2^4,"z") sage: a = z^15 sage: lift2smallest_field2(a) (1, Finite Field of size 2)
WARNING: Since coercion (the FF(b) step) has a bug in it, this *only works* in the case when you *know* F is a prime field.
Author: David Joyner
g, v) |
Returns permutation of rows g*v. Works on lists, matrices, sequences and vectors (by permuting coordinates). The code requires switching from i to i+1 (and back again) since the SymmetricGroup is, by convention, the symmetric group on the "letters" 1, 2, ..., n (not 0, 1, ..., n-1).
sage: V = VectorSpace(GF(3),5) sage: v = V([0,1,2,0,1]) sage: G = SymmetricGroup(5) sage: g = G([(1,2,3)]) sage: permutation_action(g,v) (1, 2, 0, 0, 1) sage: g = G([()]) sage: permutation_action(g,v) (0, 1, 2, 0, 1) sage: g = G([(1,2,3,4,5)]) sage: permutation_action(g,v) (1, 2, 0, 1, 0) sage: L = Sequence([1,2,3,4,5]) sage: permutation_action(g,L) [2, 3, 4, 5, 1] sage: MS = MatrixSpace(GF(3),3,7) sage: A = MS([[1,0,0,0,1,1,0],[0,1,0,1,0,1,0],[0,0,0,0,0,0,1]]) sage: S5 = SymmetricGroup(5) sage: g = S5([(1,2,3)]) sage: A [1 0 0 0 1 1 0] [0 1 0 1 0 1 0] [0 0 0 0 0 0 1] sage: permutation_action(g,A) [0 1 0 1 0 1 0] [0 0 0 0 0 0 1] [1 0 0 0 1 1 0]
It also works on lists and is a "left action":
sage: v = [0,1,2,0,1] sage: G = SymmetricGroup(5) sage: g = G([(1,2,3)]) sage: gv = permutation_action(g,v); gv [1, 2, 0, 0, 1] sage: permutation_action(g,v) == g(v) True sage: h = G([(3,4)]) sage: gv = permutation_action(g,v) sage: hgv = permutation_action(h,gv) sage: hgv == permutation_action(h*g,v) True
Author: David Joyner, licensed under the GPL v2 or greater.
m0) |
This is the generator matrix of a Walsh code. The matrix of codewords correspond to a Hadamard matrix.
sage: walsh_matrix(2) [0 0 1 1] [0 1 0 1] sage: walsh_matrix(3) [0 0 0 0 1 1 1 1] [0 0 1 1 0 0 1 1] [0 1 0 1 0 1 0 1] sage: C = LinearCode(walsh_matrix(4)); C Linear code of length 16, dimension 4 over Finite Field of size 2 sage: C.minimum_distance() 8
REFERENCES: http://en.wikipedia.org/wiki/Hadamard_matrix