Guava error-correcting code constructions.

This module only contains Guava wrappers (Guava is an optional GAP package).

AUTHORS:

  • David Joyner (2005-11-22, 2006-12-03): initial version
  • Nick Alexander (2006-12-10): factor GUAVA code to guava.py
  • David Joyner (2007-05): removed Golay codes, toric and trivial codes and placed them in code_constructions; renamed RandomLinearCode to RandomLinearCodeGuava
  • David Joyner (2008-03): removed QR, XQR, cyclic and ReedSolomon codes
  • David Joyner (2009-05): added “optional package” comments, fixed some docstrings to to be sphinx compatible

Functions

sage.coding.guava.BinaryReedMullerCode(r, k)

The binary ‘Reed-Muller code’ with dimension k and order r is a code with length 2^k and minimum distance 2^k-r (see for example, section 1.10 in [HP]). By definition, the r^{th} order binary Reed-Muller code of length n=2^m, for 0 \leq r \leq m, is the set of all vectors (f(p)\ |\ p \\in GF(2)^m), where f is a multivariate polynomial of degree at most r in m variables.

INPUT:

  • r, k – positive integers with 2^k>r.

OUTPUT:

Returns the binary ‘Reed-Muller code’ with dimension k and order r.

EXAMPLE:

sage: C = codes.BinaryReedMullerCode(2,4); C  # optional - gap_packages (Guava package)
Linear code of length 16, dimension 11 over Finite Field of size 2
sage: C.minimum_distance()              # optional - gap_packages (Guava package)
4
sage: C.gen_mat()                       # optional - gap_packages (Guava package)
[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1]
[0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1]
[0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1]
[0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1]
[0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1]
[0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1]

AUTHOR: David Joyner (11-2005)

sage.coding.guava.QuasiQuadraticResidueCode(p)

A (binary) quasi-quadratic residue code (or QQR code), as defined by Proposition 2.2 in [BM], has a generator matrix in the block form G=(Q,N). Here Q is a p \times p circulant matrix whose top row is (0,x_1,...,x_{p-1}), where x_i=1 if and only if i is a quadratic residue \mod p, and N is a p \times p circulant matrix whose top row is (0,y_1,...,y_{p-1}), where x_i+y_i=1 for all i.

INPUT:

  • p – a prime >2.

OUTPUT:

Returns a QQR code of length 2p.

EXAMPLES:

sage: C = codes.QuasiQuadraticResidueCode(11); C   # optional - gap_packages (Guava package)
Linear code of length 22, dimension 11 over Finite Field of size 2

REFERENCES:

[BM]Bazzi and Mitter, {it Some constructions of codes from group actions}, (preprint March 2003, available on Mitter’s MIT website).
[Jresidue]D. Joyner, {it On quadratic residue codes and hyperelliptic curves}, (preprint 2006)

These are self-orthogonal in general and self-dual when p \\equiv 3 \\pmod 4.

AUTHOR: David Joyner (11-2005)

sage.coding.guava.RandomLinearCodeGuava(n, k, F)

The method used is to first construct a k \times n matrix of the block form (I,A), where I is a k \times k identity matrix and A is a k \times (n-k) matrix constructed using random elements of F. Then the columns are permuted using a randomly selected element of the symmetric group S_n.

INPUT:

  • n,k – integers with n>k>1.

OUTPUT:

Returns a “random” linear code with length n, dimension k over field F.

EXAMPLES:

sage: C = codes.RandomLinearCodeGuava(30,15,GF(2)); C      # optional - gap_packages (Guava package)
Linear code of length 30, dimension 15 over Finite Field of size 2
sage: C = codes.RandomLinearCodeGuava(10,5,GF(4,'a')); C      # optional - gap_packages (Guava package)
Linear code of length 10, dimension 5 over Finite Field in a of size 2^2

AUTHOR: David Joyner (11-2005)

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