Symplectic Quasi Self-dual, Self-dual, and LCD Codes over Non-unital Rings of Order Six

TL;DR

This paper studies codes over specific non-unital rings of order six, characterizing their symplectic self-duality properties and classifying certain codes using automorphism groups and double-coset enumeration.

Contribution

It introduces a new framework for analyzing codes over non-unital rings of order six, including symplectic self-dual and LCD codes, with classification results for short even lengths.

Findings

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

Introduction of symplectic nice and LCD codes.

Classification of codes up to permutation equivalence for short even lengths.

Abstract

We consider codes over the two semi-local non-unital rings of order six, \[ H_{23} = \langle a,b \mid 2a=0, 3b = 0, a^2=a, b^2 = 0, ab = 0 = ba \rangle,\] and \[H_{32} = \langle a,b \mid 2a=0, 3b = 0, a^2=0, b^2 = b, ab = 0 = ba \rangle. \] with respect to a symplectic inner product. Via the decomposition $\mathcal{C} = a\,\mathcal{C}_a\oplus b\,\mathcal{C}_b$, any $H_z$-code splits into a binary component $\mathcal{C}_a$ and a ternary component $\mathcal{C}_b$; this yields characterizations of symplectic self-orthogonal, self-dual, and quasi self-dual (QSD) codes, and allows the introduction of symplectic nice and symplectic linear complementary dual (LCD) codes. Using the automorphism groups of $\mathcal{C}_a$ and $\mathcal{C}_b$ and double-coset enumeration, we classify, up to permutation equivalence, all symplectic self-orthogonal and QSD $H_z$-codes of short even lengths.