Symplectic Hulls over a Non-Unital Ring

TL;DR

This paper investigates symplectic hulls over a specific non-unital ring, characterizing their structure, exploring their properties under code operations, and classifying optimal codes for small lengths.

Contribution

It provides a detailed analysis of symplectic hulls over a non-unital ring, including generator matrices, code sum behavior, construction techniques, and classification of optimal codes.

Findings

Characterization of residue and torsion codes of symplectic hulls

Construction techniques for extending codes with desired hull properties

Classification of optimal codes for small lengths

Abstract

This paper presents the study of the symplectic hulls over a non-unital ring $ E= \langle \kappa,\tau \mid 2 \kappa =2 \tau=0,~ \kappa^2=\kappa,~ \tau^2=\tau,~ \kappa \tau=\kappa,~ \tau \kappa=\tau \rangle$. We first identify the residue and torsion codes of the left, right, and two-sided symplectic hulls, and characterize the generator matrix of the two-sided symplectic hull of a free $E$-linear code. Then, we explore the symplectic hull of the sum of two free $E$-linear codes. Subsequently, we provide two build-up techniques that extend a free $E$-linear code of smaller length and symplectic hull-rank to one of larger length and symplectic hull-rank. Further, for free $E$-linear codes, we discuss the permutation equivalence and investigate the symplectic hull-variation problem. An application of this study is given by classifying the free $E$-linear optimal codes for smaller lengths.