Communicated by J. Wolfmann.
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.