Quasi-abelian codes

Abstract

We study $$H$$ H -quasi-abelian codes in $$\mathbb F _q[G]$$ F q [ G ] , where $$H\le G$$ H ≤ G are abelian groups such that $$\gcd (|H|,q)=1$$ gcd ( | H | , q ) = 1 . Such codes are generalizations of quasi-cyclic codes and can be viewed as linear codes over the group ring $$\mathbb F _q[H]$$ F q [ H ] . Using the Discrete Fourier Transform, $$\mathbb F _q[H]$$ 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 $$q$$ q -cyclotomic classes of $$H$$ H . In particular, when $$H$$ H is an elementary $$p$$ p -group, we characterize the $$q$$ q -cyclotomic classes of $$H$$ H and give an explicit formula for the number of self-dual $$H$$ 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.