Constructions of generator matrices using the GUAVA package for GAP¶

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 be sphinx compatible
Dima Pasechnik (2019-11): port to libgap

sage.coding.guava.QuasiQuadraticResidueCode(p)[source]¶

A (binary) quasi-quadratic residue code (or QQR code).

Follows the definition of Proposition 2.2 in [BM2003]. The code 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: a QQR code of length \(2p\)

EXAMPLES:

Sage

sage: C = codes.QuasiQuadraticResidueCode(11); C   # optional - gap_package_guava
[22, 11] linear code over GF(2)

Python

>>> from sage.all import *
>>> C = codes.QuasiQuadraticResidueCode(Integer(11)); C   # optional - gap_package_guava
[22, 11] linear code over GF(2)

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)[source]¶

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: a “random” linear code with length \(n\), dimension \(k\) over field \(F\)

EXAMPLES:

Sage

sage: C = codes.RandomLinearCodeGuava(30,15,GF(2)); C      # optional - gap_package_guava
[30, 15] linear code over GF(2)
sage: C = codes.RandomLinearCodeGuava(10,5,GF(4,'a')); C   # optional - gap_package_guava
[10, 5] linear code over GF(4)

Python

>>> from sage.all import *
>>> C = codes.RandomLinearCodeGuava(Integer(30),Integer(15),GF(Integer(2))); C      # optional - gap_package_guava
[30, 15] linear code over GF(2)
>>> C = codes.RandomLinearCodeGuava(Integer(10),Integer(5),GF(Integer(4),'a')); C   # optional - gap_package_guava
[10, 5] linear code over GF(4)

AUTHOR: David Joyner (11-2005)