Hulls of Free Linear Codes over a Non-Unital Ring

TL;DR

This paper explores the structure and construction of hull codes of free linear codes over a specific non-unital ring, providing explicit formulas, construction methods, and classification results for codes up to length 8.

Contribution

It introduces explicit generator matrices for hulls, four new construction methods, and classifies optimal codes over a non-unital ring for lengths up to 8.

Findings

Explicit form of hull generator matrices

Four construction methods for larger codes

Classification of optimal codes up to length 8

Abstract

This paper investigates the hull codes of free linear codes 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$. Initially, we examine the residue and torsion codes of various hulls of $E$-linear codes and obtain an explicit form of the generator matrix of the hull of a free $E$-linear code. Then, we propose four build-up construction methods to construct codes with a larger length and hull-rank from codes with a smaller length and hull-rank. Some illustrative examples are also given to support our build-up construction methods. Subsequently, we study the permutation equivalence of two free $E$-linear codes and discuss the hull-variation problem. As an application, we classify optimal free $E$-linear codes for lengths up to $8$.