Hulls of Free Linear Codes over a Non-Unital Ring

Anup Kushwaha; Om Prakash

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arXiv · 2512.12335

Hulls of Free Linear Codes over a Non-Unital Ring

Anup Kushwaha·Om Prakash

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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 $8$.

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Anup Kushwaha, Om Prakash. 2025-12-13. Hulls of Free Linear Codes over a Non-Unital Ring. https://arxiv.org/abs/2512.12335

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