On equivalent classes of minimal Abelian codes

Abstract

Let Fq be a finite field and G a finite abelian group. An abelian code is an ideal of Fq[G]. Two abelian codes c1 and c2 of Fq[G] are equivalent if there exists an automorphism of G whose linear extension to Fq[G] maps c1 onto c2. MacWilliams determined the number of equivalent classes of minimal abelian codes (minimal ideals) in F2[G] for cyclic group G of odd cardinality. Miller claimed that MacWilliams' result remains true in general, which is however not correct as pointed out by Ferraz, Guerreiro and Polcino Milies. In this paper we completely determine the number of equivalent classes of minimal abelian codes for any Fq[G].