Quasi-abelian codes

Abstract

We study $H$-quasi-abelian codes in $\mathbb F _q[G]$, where $H\le G$ are abelian groups such that $\gcd (|H|,q)=1$. Such codes are generalizations of quasi-cyclic codes and can be viewed as linear codes over the group ring $\mathbb F _q[H]$. Using the Discrete Fourier Transform, $\mathbb F _q[H]$ can be decomposed as a direct product of finite fields. This decomposition leads us to a structural characterization of quasi-abelian codes and their duals. Necessary and sufficient conditions for such codes to be self-dual are given together with the enumeration based on $q$-cyclotomic classes of $H$. In particular, when $H$ is an elementary $p$-group, we characterize the $q$-cyclotomic classes of $H$ and give an explicit formula for the number of self-dual $H$-quasi abelian codes. Analogous to 1-generator quasi-cyclic codes, we investigate the structural characterization and enumeration of 1-generator quasi-abelian codes. We show that the class of binary self-dual (strictly) quasi-abelian codes is asymptotically good. Finally, we present four strictly quasi-abelian codes and ten codes obtained by puncturing and shortening of these codes, whose minimum distances are better than the lower bound in Grassl’s online table.