Hermitian LCD 2-Quasi Abelian Codes over Finite Chain Rings
🤖 AI Summary
This study investigates the existence and asymptotic performance of Hermitian complementary dual (LCD) 2-quasi-abelian codes over finite fields and finite chain rings. By leveraging representation theory over finite rings, structural analysis of Hermitian duality, and algebraic code construction techniques, this work extends Hermitian LCD 2-quasi-abelian codes for the first time to the setting of finite chain rings and characterizes their algebraic structure. The main contributions include establishing the existence of asymptotically good constructions of such codes over both finite fields and finite chain rings, as well as providing a complete classification in the regime of small relative minimum distance. These results significantly broaden the applicability of classical coding theory beyond field-based structures to more general algebraic settings.
Application Category
📝 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 can achieve 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.
Problem
Research questions and friction points this paper is trying to address.
Hermitian LCD codes
2-quasi-abelian codes
finite chain rings
asymptotically good codes
relative minimum weight
Innovation
Methods, ideas, or system contributions that make the work stand out.
Hermitian LCD codes
2-quasi-abelian codes
finite chain rings
asymptotically good codes
relative minimum weight
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