This file contains constructions of error-correcting codes which are pure Python/Sage and not obtained from wrapping GUAVA functions. The GUAVA wrappers are in guava.py.
All codes available here can be accessed through the codes object:
sage: codes.HammingCode(3,GF(2))
Linear code of length 7, dimension 4 over Finite Field of size 2
Let \(F\) be a finite field with \(q\) elements. Here’s a constructive definition of a cyclic code of length \(n\).
The polynomial notation for the code is to call \(c_0+c_1x+...+c_{n-1}x^{n-1}\) the codeword (instead of \((c_0,c_1,...,c_{n-1})\)). The polynomial \(h(x)=(x^n-1)/g(x)\) is called the check polynomial of \(C\).
Let \(n\) be a positive integer relatively prime to \(q\) and let \(\alpha\) be a primitive \(n\)-th root of unity. Each generator polynomial \(g\) of a cyclic code \(C\) of length \(n\) has a factorization of the form
where \(\{k_1,...,k_r\} \subset \{0,...,n-1\}\). The numbers \(\alpha^{k_i}\), \(1 \leq i \leq r\), are called the zeros of the code \(C\). Many families of cyclic codes (such as BCH codes and the quadratic residue codes) are defined using properties of the zeros of \(C\).
REFERENCES:
[HP] | (1, 2, 3, 4, 5, 6, 7, 8, 9) W. C. Huffman, V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, 2003. |
AUTHOR:
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) \(Z = \{a^{b},a^{b+1}, ..., a^{b+delta-2}\}\), where a is a primitive \(n^{th}\) root of unity in the splitting field \(GF(q^m)\), b is an integer \(0\leq b\leq n-delta+1\) and m is the multiplicative order of q modulo n. (The integers \(b,...,b+delta-2\) typically lie in the range \(1,...,n-1\).) The integer \(delta \geq 1\) 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 \(Z\) above.
Special cases are b=1 (resulting codes are called ‘narrow-sense’ BCH codes), and \(n=q^m-1\) (known as ‘primitive’ BCH codes).
It may happen that several values of delta give rise to the same BCH code. The largest 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 \(d\geq delta\).
EXAMPLES:
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 = codes.CyclicCode(8,g); C
Linear code of length 8, dimension 4 over Finite Field of size 3
sage: C.minimum_distance()
4
sage: C = codes.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 = codes.BCHCode(8,3,GF(3)); C
Linear code of length 8, dimension 5 over Finite Field of size 3
sage: C.minimum_distance()
3
sage: C = codes.BCHCode(26, 5, GF(5), b=1); C
Linear code of length 26, dimension 10 over Finite Field of size 5
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 \(g(x)=1+x^2+x^4+x^5+x^6+x^{10}+x^{11}\). Extending it yields the extended Golay code (see ExtendedBinaryGolayCode).
EXAMPLE:
sage: C = codes.BinaryGolayCode()
sage: C
Linear code of length 23, dimension 12 over Finite Field of size 2
sage: C.minimum_distance()
7
sage: C.minimum_distance(algorithm='gap') # long time, check d=7
7
AUTHORS:
If g is a polynomial over GF(q) which divides \(x^n-1\) then this constructs the code “generated by g” (ie, the code associated with the principle ideal \(gR\) in the ring \(R = GF(q)[x]/(x^n-1)\) 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 assumption in the theory of cyclic codes), and (b) \(g\) must divide \(x^n-1\). If ignore=True, instead of returning an error, a code generated by \(gcd(x^n-1,g)\) is created.
EXAMPLES:
sage: P.<x> = PolynomialRing(GF(3),"x")
sage: g = x-1
sage: C = codes.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 = codes.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 = codes.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 = codes.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: “\(x\) must divide \(x^4 - 1\)”. 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 = codes.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 “\(x^2 + 2\) must divide \(x^5 - 1\)”.
If h is a polynomial over GF(q) which divides \(x^n-1\) then this constructs the code “generated by \(g = (x^n-1)/h\)” (ie, the code associated with the principle ideal \(gR\) in the ring \(R = GF(q)[x]/(x^n-1)\) in the usual way). The option “ignore” says to ignore the condition that the characteristic of the base field does not divide the length (the usual assumption in the theory of cyclic codes).
EXAMPLES:
sage: P.<x> = PolynomialRing(GF(3),"x")
sage: C = codes.CyclicCodeFromCheckPolynomial(4,x + 1); C
Linear code of length 4, dimension 1 over Finite Field of size 3
sage: C = codes.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]
If g is a polynomial over GF(q) which divides \(x^n-1\) then this constructs the code “generated by g” (ie, the code associated with the principle ideal \(gR\) in the ring \(R = GF(q)[x]/(x^n-1)\) 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 assumption in the theory of cyclic codes), and (b) \(g\) must divide \(x^n-1\). If ignore=True, instead of returning an error, a code generated by \(gcd(x^n-1,g)\) is created.
EXAMPLES:
sage: P.<x> = PolynomialRing(GF(3),"x")
sage: g = x-1
sage: C = codes.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 = codes.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 = codes.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 = codes.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: “\(x\) must divide \(x^4 - 1\)”. 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 = codes.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 “\(x^2 + 2\) must divide \(x^5 - 1\)”.
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.
EXAMPLES:
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: codes.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)
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.
EXAMPLES:
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: codes.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.
EXAMPLES:
sage: C = codes.ExtendedBinaryGolayCode()
sage: C
Linear code of length 24, dimension 12 over Finite Field of size 2
sage: C.minimum_distance()
8
sage: C.minimum_distance(algorithm='gap') # long time, check d=8
8
AUTHORS:
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], Section 6.6.3-6.6.4.)
INPUT:
OUTPUT: Returns an extended quadratic residue code.
EXAMPLES:
sage: C1 = codes.QuadraticResidueCode(7,GF(2))
sage: C2 = C1.extended_code()
sage: C3 = codes.ExtendedQuadraticResidueCode(7,GF(2)); C3
Linear code of length 8, dimension 4 over Finite Field of size 2
sage: C2 == C3
True
sage: C = codes.ExtendedQuadraticResidueCode(17,GF(2))
sage: C
Linear code of length 18, dimension 9 over Finite Field of size 2
sage: C3 = codes.QuadraticResidueCodeOddPair(7,GF(2))[0]
sage: C3x = C3.extended_code()
sage: C4 = codes.ExtendedQuadraticResidueCode(7,GF(2))
sage: C3x == C4
True
AUTHORS:
ExtendedTernaryGolayCode returns a ternary Golay code. This is a self-dual perfect [12,6,6] code.
EXAMPLES:
sage: C = codes.ExtendedTernaryGolayCode()
sage: C
Linear code of length 12, dimension 6 over Finite Field of size 3
sage: C.minimum_distance()
6
sage: C.minimum_distance(algorithm='gap') # long time, check d=6
6
AUTHORS:
Implements the Hamming codes.
The \(r^{th}\) Hamming code over \(F=GF(q)\) is an \([n,k,d]\) code with length \(n=(q^r-1)/(q-1)\), dimension \(k=(q^r-1)/(q-1) - r\) and minimum distance \(d=3\). 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:
OUTPUT: Returns the r-th q-ary Hamming code.
EXAMPLES:
sage: codes.HammingCode(3,GF(2))
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: C = codes.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.minimum_distance(algorithm='gap') # long time, check d=3
3
sage: C = codes.HammingCode(3,GF(4,'a')); C
Linear code of length 21, dimension 18 over Finite Field in a of size 2^2
While the codes object now gathers all code constructors, HammingCode is still available in the global namespace:
sage: HammingCode(3,GF(2))
doctest:1: DeprecationWarning: This method soon will not be available in that way anymore. To use it, you can now call it by typing codes.HammingCode
See http://trac.sagemath.org/15445 for details.
Linear code of length 7, dimension 4 over Finite Field of size 2
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
(“Short exact” means (a) the arrow \(G\) is injective, i.e., \(G\) is a full-rank \(k\times n\) matrix, (b) the arrow \(H\) is surjective, and (c) \({\rm image}(G)={\rm kernel}(H)\).)
EXAMPLES:
sage: C = codes.HammingCode(3,GF(2))
sage: H = C.check_mat(); H
[1 0 1 0 1 0 1]
[0 1 1 0 0 1 1]
[0 0 0 1 1 1 1]
sage: codes.LinearCodeFromCheckMatrix(H) == C
True
sage: C = codes.HammingCode(2,GF(3))
sage: H = C.check_mat(); H
[1 0 1 1]
[0 1 1 2]
sage: codes.LinearCodeFromCheckMatrix(H) == C
True
sage: C = codes.RandomLinearCode(10,5,GF(4,"a"))
sage: H = C.check_mat()
sage: codes.LinearCodeFromCheckMatrix(H) == C
True
A quadratic residue code (or QR code) is a cyclic code whose generator polynomial is the product of the polynomials \(x-\alpha^i\) (\(\alpha\) is a primitive \(n^{th}\) root of unity; \(i\) ranges over the set of quadratic residues modulo \(n\)).
See QuadraticResidueCodeEvenPair and QuadraticResidueCodeOddPair for a more general construction.
INPUT:
OUTPUT: Returns a quadratic residue code.
EXAMPLES:
sage: C = codes.QuadraticResidueCode(7,GF(2))
sage: C
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: C = codes.QuadraticResidueCode(17,GF(2))
sage: C
Linear code of length 17, dimension 9 over Finite Field of size 2
sage: C1 = codes.QuadraticResidueCodeOddPair(7,GF(2))[0]
sage: C2 = codes.QuadraticResidueCode(7,GF(2))
sage: C1 == C2
True
sage: C1 = codes.QuadraticResidueCodeOddPair(17,GF(2))[0]
sage: C2 = codes.QuadraticResidueCode(17,GF(2))
sage: C1 == C2
True
AUTHORS:
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.
EXAMPLES:
sage: codes.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: codes.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: codes.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 = codes.QuadraticResidueCodeEvenPair(7,GF(2))[0]
sage: C1.is_self_orthogonal()
True
sage: C2 = codes.QuadraticResidueCodeEvenPair(7,GF(2))[1]
sage: C2.is_self_orthogonal()
True
sage: C3 = codes.QuadraticResidueCodeOddPair(17,GF(2))[0]
sage: C4 = codes.QuadraticResidueCodeEvenPair(17,GF(2))[1]
sage: C3 == C4.dual_code()
True
This is consistent with Theorem 6.6.9 and Exercise 365 in [HP].
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.
EXAMPLES:
sage: codes.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: codes.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: codes.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 = codes.QuadraticResidueCodeOddPair(17,GF(2))[1]
sage: C1x = C1.extended_code()
sage: C2 = codes.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 = codes.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].
The method used is to first construct a \(k \times n\) 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 \(n>k\), and a finite field F
OUTPUT: Returns a “random” linear code with length n, dimension k over field F.
EXAMPLES:
sage: C = codes.RandomLinearCode(30,15,GF(2))
sage: C
Linear code of length 30, dimension 15 over Finite Field of size 2
sage: C = codes.RandomLinearCode(10,5,GF(4,'a'))
sage: C
Linear code of length 10, dimension 5 over Finite Field in a of size 2^2
AUTHORS:
Given a finite field \(F\) of order \(q\), let \(n\) and \(k\) be chosen such that \(1 \leq k \leq n \leq q\). Pick \(n\) distinct elements of \(F\), denoted \(\{ x_1, x_2, ... , x_n \}\). Then, the codewords are obtained by evaluating every polynomial in \(F[x]\) of degree less than \(k\) at each \(x_i\):
\(C\) is a \([n, k, n-k+1]\) code. (In particular, \(C\) is MDS.)
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)
EXAMPLES:
sage: C = codes.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 = codes.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: C.minimum_distance(algorithm='gap') # long time, check d=n-k+1
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 = codes.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
While the codes object now gathers all code constructors, ReedSolomonCode is still available in the global namespace:
sage: ReedSolomonCode(6,4,GF(7))
doctest:1: DeprecationWarning: This method soon will not be available in that way anymore. To use it, you can now call it by typing codes.ReedSolomonCode
See http://trac.sagemath.org/15445 for details.
Linear code of length 6, dimension 4 over Finite Field of size 7
REFERENCES:
TernaryGolayCode returns a ternary Golay code. This is a perfect [11,6,5] code. It is also equivalent to a cyclic code, with generator polynomial \(g(x)=2+x^2+2x^3+x^4+x^5\).
EXAMPLES:
sage: C = codes.TernaryGolayCode()
sage: C
Linear code of length 11, dimension 6 over Finite Field of size 3
sage: C.minimum_distance()
5
sage: C.minimum_distance(algorithm='gap') # long time, check d=5
5
AUTHORS:
Let \(P\) denote a list of lattice points in \(\ZZ^d\) and let \(T\) denote the set of all points in \((F^x)^d\) (ordered in some fixed way). Put \(n=|T|\) and let \(k\) denote the dimension of the vector space of functions \(V = \mathrm{Span}\{x^e \ |\ e \in P\}\). The associated toric code \(C\) is the evaluation code which is the image of the evaluation map
where \(x^e\) is the multi-index notation (\(x=(x_1,...,x_d)\), \(e=(e_1,...,e_d)\), and \(x^e = x_1^{e_1}...x_d^{e_d}\)), where \(eval_T (f(x)) = (f(t_1),...,f(t_n))\), and where \(T=\{t_1,...,t_n\}\). This function returns the toric codes discussed in [J].
INPUT:
OUTPUT: Returns toric code with length n = , dimension k over field F.
EXAMPLES:
sage: C = codes.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 = codes.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 = codes.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:
REFERENCES:
[J] | D. Joyner, Toric codes over finite fields, Applicable Algebra in Engineering, Communication and Computing, 15, (2004), p. 63-79. |
x.__init__(...) initializes x; see help(type(x)) for signature
Returns the binary Walsh code of length \(2^m\). The matrix of codewords correspond to a Hadamard matrix. This is a (constant rate) binary linear \([2^m,m,2^{m-1}]\) code.
EXAMPLES:
sage: C = codes.WalshCode(4); C
Linear code of length 16, dimension 4 over Finite Field of size 2
sage: C = codes.WalshCode(3); C
Linear code of length 8, dimension 3 over Finite Field of size 2
sage: C.spectrum()
[1, 0, 0, 0, 7, 0, 0, 0, 0]
sage: C.minimum_distance()
4
sage: C.minimum_distance(algorithm='gap') # check d=2^(m-1)
4
REFERENCES:
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 is not None, the q-cyclotomic coset mod n containing t)
Let q, n be relatively print positive integers and let \(A = q^{ZZ}\). 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 \(z\) of \(GF(q^k)\). The minimal polynomial of \(z^s\) over \(GF(q)\) is given by
where \(C_s\) is the q-cyclotomic coset mod n containing s, \(n = q^k - 1\).
EXAMPLES:
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 looks strange but is correct, since the elements of the cosets are in ZZ/nZZ and 2 = 0 in ZZ/2ZZ.
INPUT: S1, S2 are disjoint sublists partitioning [1, 2, ..., n-1] n1 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 b1 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 \(Q = GF(q)[x]/(x^n-1)\). For
EXAMPLES:
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 = codes.CyclicCodeFromGeneratingPolynomial(n,i1)
sage: C2 = codes.CyclicCodeFromGeneratingPolynomial(n,1-i2)
sage: C1.dual_code() == C2
True
This is a special case of Theorem 6.4.3 in [HP].
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.
EXAMPLES:
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)
AUTHORS:
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.
EXAMPLES:
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.
AUTHORS:
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).
EXAMPLES:
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
AUTHORS:
This is the generator matrix of a Walsh code. The matrix of codewords correspond to a Hadamard matrix.
EXAMPLES:
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.spectrum()
[1, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0]
This last code has minimum distance 8.
REFERENCES: