Self-dual $2$-quasi-abelian Codes

TL;DR

This paper introduces and analyzes self-dual and self-orthogonal quasi-abelian codes of index 2 over finite fields, proving their asymptotic goodness under certain conditions and counting their quantities.

Contribution

It defines new classes of self-dual and self-orthogonal quasi-abelian codes of index 2 and proves their asymptotic goodness, expanding coding theory knowledge.

Findings

Self-dual quasi-abelian codes are asymptotically good when -1 is a square in the field.

The number of such codes can be explicitly counted.

Self-orthogonal quasi-abelian codes of index 2 always exist and are asymptotically good.

Abstract

A kind of self-dual quasi-abelian codes of index $2$ over any finite field $F$ is introduced. By counting the number of such codes and the number of the codes of this kind whose relative minimum weights are small, such codes are proved to be asymptotically good provided $-1$ is a square in $F$. Moreover, a kind of self-orthogonal quasi-abelian codes of index $2$ are defined; and such codes always exist. In a way similar to that for self-dual quasi-abelian codes of index $2$, it is proved that the kind of the self-orthogonal quasi-abelian codes of index $2$ is asymptotically good.