Duality of Codes over Non-unital Rings of Order Six

TL;DR

This paper explores the duality properties of codes over two non-unital rings of order six, establishing their relationships with binary and ternary codes and introducing new code constructions.

Contribution

It introduces a comprehensive theory on duality of codes over specific non-unital rings, including classifications, constructions, and properties of various code types.

Findings

Characterization of self-orthogonal, self-dual, and QSD codes over the rings

Development of a building-up construction for self-orthogonal codes

Classification of self-orthogonal codes for short lengths

Abstract

We present some basic theory on the duality of codes over two non-unital rings of order $6$, namely $H_{23}$ and $H_{32}$. For a code $\mathcal{C}$ over these rings, we associate a binary code $\mathcal{C}_a$ and a ternary code $\mathcal{C}_b$. We characterize self-orthogonal, self-dual and quasi self-dual (QSD) codes over these rings using the codes $\mathcal{C}_a$ and $\mathcal{C}_b$. In addition, we present a building-up construction for self-orthogonal codes, introduce cyclic codes and linear complementary dual (LCD) codes. We also gave a classification of self-orthogonal codes for short lengths.