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.

翻译：本文引入了一类有限域上的Hermitian LCD 2-拟阿贝尔码，并对其中相对最小权值较小的码进行了全面的枚举。我们证明了此类码在有限域上是渐近好的。此外，我们将分析推广到有限链环，通过刻画该背景下的2-拟阿贝尔码，并证明了有限链环上同样存在渐近好的Hermitian LCD 2-拟阿贝尔码。