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.
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)