# 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
sage.coding.guava.QuasiQuadraticResidueCode(p)

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:

Returns a QQR code of length $$2p$$.

EXAMPLES:

sage: C = codes.QuasiQuadraticResidueCode(11); C   # optional - gap_packages (Guava package)
[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)

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


AUTHOR: David Joyner (11-2005)