Hermitian LCD $2$-Quasi Abelian Codes over Finite Chain Rings

TL;DR

This paper introduces and characterizes Hermitian LCD 2-quasi-abelian codes over finite fields and chain rings, demonstrating their asymptotic goodness and providing enumeration based on minimum weights.

Contribution

It extends the theory of Hermitian LCD codes to 2-quasi-abelian codes over finite fields and chain rings, including enumeration and asymptotic properties.

Findings

Codes are asymptotically good over finite fields.

Enumeration of codes with small relative minimum weights.

Existence of asymptotically good codes over finite chain rings.

Abstract

This paper introduces a class of Hermitian LCD $2$-quasi-abelian codes over finite fields and presents a comprehensive enumeration of these codes in which relative minimum weights are small. We show that such codes are asymptotically good over finite fields. Furthermore, we extend our analysis to finite chain rings by characterizing $2$-quasi-abelian codes in this setting and proving the existence of asymptotically good Hermitian LCD $2$-quasi-abelian codes over finite chain rings as well.