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

TL;DR

The work addresses constructing Hermitian LCD $2$-quasi-abelian codes over finite fields and finite chain rings that are asymptotically good. It develops a parametric family $\mathcal{C}_{a,b}$ in the group ring $\mathbb{F}G$ with a Hermitian form $\langle\cdot,\cdot\rangle_{\sigma}$ and identifies a central class $\mathcal{C}_{1,\beta}$ determined by $\beta\beta^{\tau}=\lambda-1$, proving these codes are Hermitian LCD when $\lambda\in\mathbb{F}^{\times}$. The paper counts the family $\mathcal{D}_{\lambda}$ of such codes, derives explicit size formulas, and uses the $q$-ary entropy function $h_q(\delta)$ to show a positive fraction attain relative distance exceeding a given $\delta$, establishing asymptotic goodness with rate $1/2$. It then lifts the field results to finite chain rings, proving that LCD properties are preserved through projection and radix-based lifting, and concluding the existence of asymptotically good Hermitian LCD $2$-quasi-abelian codes over finite chain rings as well. Overall, the results extend LCD-code theory to a structured class of abelian-group codes with rigorous asymptotic guarantees and practical lifting techniques for ring-valued codes.

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.

Theorems & Definitions (55)

• Lemma 2.1
• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5