Duality of Codes over Non-unital Rings of Order Six

TL;DR

This work extends coding theory to two non-unital rings of order six, $H_{23}$ and $H_{32}$, by decomposing codes as $\mathcal{C}=a\mathcal{C}_a\oplus b\mathcal{C}_b$ with $\mathcal{C}_a$ binary and $\mathcal{C}_b$ ternary. It derives duality relationships, providing necessary and sufficient conditions for self-orthogonality, self-duality, QSD, LCD, and cyclicity in terms of the component codes, and establishes building-up constructions to obtain longer self-orthogonal codes. The paper also surveys cyclic and LCD criteria and delivers extensive numerical classifications of self-orthogonal codes up to length $7$ over both rings, using MAGMA, including notable QSD families for $H_{23}$ and a precise count for $H_{32}$. By linking codes over $H_{23}$ and $H_{32}$ to binary/ternary codes, it reveals rich structure and practical construction methods with potential implications for code design over non-unital algebras.

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.

Theorems & Definitions (20)

• Theorem 4.1
• Theorem 4.2
• Corollary 4.3
• Theorem 4.4
• Theorem 4.5
• Corollary 4.6
• Theorem 4.7
• Proposition 4.8
• Theorem 4.9
• Corollary 4.10