Minimal group codes over alternating groups

In this work we show that every minimal code in a semisimple group algebra $\mathbb{F}_qG$ is essential if $G$ is a simple group. Since the alternating group $A_n$ is simple if $n=3$ or $n\geq 5$, we present some examples of minimal codes in $\mathbb{F}_qA_n$. For this purpose, if $char(\mathbb{F}_q)> n$, we present the Wedderburn-Artin decomposition of $\mathbb{F}_qS_n$ and $\mathbb{F}_qA_n$ and explicit some of the centrally primitive idempotents of $\mathbb{F}_qS_n$ and $\mathbb{F}_qA_n$.Comment: 16 page

On minimal easily computable dimension group algebras, and group codes

Finite semisimple group algebras for which all the minimal ideals are easily computable dimension (ECD) are characterized and some lower bounds for the minimum Hamming distance of group codes in these algebras are offered. Examples illustrating the main results are provided

Semisimple group codes and dihedral codes

We consider codes that are given as two-sided ideals in a semisimple finite group algebra FqG defined by idempotents constructed from subgroups of G in a natural way and compute their dimensions and weights. We give a criterion to decide when these ideals are all the minimal two-sided ideals o f FqG in the case when G is a dihedral group and extend these results also to a family of quaternion group codes. In the final sectio n, we give a method of decoding; i.e., of finding and correcting eve ntual transmission errors

Essential Idempotents in Group Algebras and Minimal Cyclic Codes

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Semisimple metacyclic group algebras

Given a group G of order p1 p2, where p1, p2 are primes, and Fq, a finite field of order q coprime to p1 p2, the object of this paper is to compute a complete set of primitive central idempotents of the semisimple group algebra Fq [G]. As a consequence, we obtain the structure of Fq [G] and its group of automorphisms