FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 2, PAGES 75-85

**When are all group codes of a noncommutative group Abelian
(a computational approach)?**

C. García Pillado

S. González

V. T. Markov

C. Martínez

A. A. Nechaev

Abstract

View as HTML
View as gif image

Let $G$ be
a finite group and $F$ be a field.
Any linear code over $F$ that is permutation
equivalent to some code defined by an ideal of the group ring
$FG$ will be
called a $G$-code.
The theory of these "abstract" group codes was developed in 2009.
A code is called Abelian if it is an $A$-code for some Abelian
group $A$.
Some conditions were given that all $G$-codes for some
group $G$
are Abelian but no examples of non-Abelian group codes were known at
that time.
We use a computer algebra system GAP to show that all $G$-codes over any field are
Abelian if $|G|\; <\; 128$ and $|G|$Ï
{24,48,54,60,64,72,96,108,120}, but for
$F=$**F**_{5} and
$G\; =\; S$_{4}
there exist non-Abelian $G$-codes
over $F$.
It is also shown that the existence of left non-Abelian group codes
for a given group depends in general on the field of
coefficients, while for (two-sided) group codes the corresponding
question remains open.

Location: http://mech.math.msu.su/~fpm/eng/k1112/k112/k11202h.htm

Last modified: March 6, 2012