Conference abstracts

Session S05 - Rings and Algebras

No date set

## Abelian Group Codes

### Universidade Federal da Bahia , Salvador   -   argentina.ale@gmail.com

Let $F$ be a finite field and $n$, a non negative integer. A linear code $C$ of length $n$ is a subspace of $F^n$. A (left) group code of length $n$ is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism $FG \rightarrow F^n$ for any $G$, a finite group with $\vert G \vert=n$. In this case $C$, is a (left) $G$-code. In [1], Bernal, del R\'io and Sim\'on obtain a criterion to decide when a linear code is a group code in terms of the group of permutation automorphisms of $C$, $PAut(C)$. Sabin and Lomonaco, in [4], have proved that if $C$ a $G$-code with $G$ a semidirect product of cyclic groups, then $C$ is an abelian group code. As an application of criterion and extending the result of Sabin and Lomonaco, in [1], they provide a family of groups for which every two-sided group code is an abelian group code. Pillado, González, Martínez, Markov e Nechaev describe some classes of groups and fields for which all group codes are abelian in [2]. Motivated by [3], they have shown that there exist a non-Abelian $G$-code over $F$. In order to extend the result on groups with abelian decompotition, we explore some conditions to determine a group $G$ which can written as a product of abelian subgroups, such that the $G$-codes with $G \in \mathcal{G}$ will be abelian group code.

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[1] \textsc{J. J. Bernal, Á. del Río and J.J. Simón, An intrinsical description of group of codes, \textit{Des. Codes Cryptogr. } \textbf{51}(3) 289-300 (2009).}\\

[2] \textsc{ C. García Pillado, S. González, C. Martínez, V. Markov and A. Nechaev, Group codes over non-abelian groups, \textit{ J. Algebra Appl.} \textbf{ 12 }(7) (2013).} \\

[3] \textsc{C. García Pillado, S. González, C. Martínez, V. Markov and A. Nechaev, When all group codes of a noncommutative group are groups abelian (a computational approach)?, \textit{ J. Math. Sci.} \textbf{186}(5) 578-585 (2012).} \\

[4] \textsc{ R.E. Sabin and S.J. Lomonaco}, \textit{Metacyclic Error-Correcting Codes}, AAECC, 6, 191-210 (1995).

Joint work with Thierry Petit Lob\~ao (Universidade Federal da Bahia, Brasil).