The Concatenated Structure of Quasi-Abelian Codes

TL;DR

This paper introduces a concatenated decomposition for quasi-abelian codes, establishing their equivalence with existing decompositions, and uses this to derive bounds, construct optimal codes, and analyze their asymptotic properties.

Contribution

It presents a new concatenated decomposition for quasi-abelian codes, linking it to existing decompositions and enabling new bounds and constructions.

Findings

Derived a general minimum distance bound for quasi-abelian codes

Constructed some optimal quasi-abelian codes with sharp bounds

Showed that strictly quasi-abelian linear complementary dual codes are asymptotically good

Abstract

The decomposition of a quasi-abelian code into shorter linear codes over larger alphabets was given in (Jitman, Ling, (2015)), extending the analogous Chinese remainder decomposition of quasi-cyclic codes (Ling, Sol\'e, (2001)). We give a concatenated decomposition of quasi-abelian codes and show, as in the quasi-cyclic case, that the two decompositions are equivalent. The concatenated decomposition allows us to give a general minimum distance bound for quasi-abelian codes and to construct some optimal codes. Moreover, we show by examples that the minimum distance bound is sharp in some cases. In addition, examples of large strictly quasi-abelian codes of about a half rate are given. The concatenated structure also enables us to conclude that strictly quasi-abelian linear complementary dual codes over any finite field are asymptotically good.