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.
KeywordsQuasi-abelian codes Group algebras Discrete Fourier transform Self-dual codes 1-Generator quasi-abelian codes
Mathematics Subject Classification94B15 94B60 16A26
The authors would like to thank the anonymous referees for helpful comments and suggestions. This research was supported by the National Research Foundation of Singapore under Research Grant NRF-CRP2-2007-03.
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