Primitive idempotents in group algebras and minimal abelian codes

Abstract

In this paper, the minimal abelian codes for several classes of non-cyclic abelian groups are constructed by explicitly determining a complete set of primitive idempotents in the corresponding group algebras. Furthermore, we acquire the minimum Hamming distances and the dimensions of these minimal abelian codes in the group algebra ${\mathbb {F}}_q(G_{\ell _1} \, \times \, G_{\ell _2})$, where $G_{\ell _1} \, \times \, G_{\ell _2}$ is the direct product of two cyclic groups and $\ell _1, \, \ell _2$ are prime divisors of $q-1$.