Hulls of Free Linear Codes over a Non-Unital Ring
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912792842338304 |
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| author | Kushwaha, Anup Prakash, Om |
| author_facet | Kushwaha, Anup Prakash, Om |
| contents | This paper investigates the hull codes of free linear codes over a non-unital ring $ E= \langle κ,τ\mid 2 κ=2 τ=0,~ κ^2=κ,~ τ^2=τ,~ κτ=κ,~ τκ=τ\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 $8$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12335 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hulls of Free Linear Codes over a Non-Unital Ring Kushwaha, Anup Prakash, Om Information Theory 16L30, 94B05 This paper investigates the hull codes of free linear codes over a non-unital ring $ E= \langle κ,τ\mid 2 κ=2 τ=0,~ κ^2=κ,~ τ^2=τ,~ κτ=κ,~ τκ=τ\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 $8$. |
| title | Hulls of Free Linear Codes over a Non-Unital Ring |
| topic | Information Theory 16L30, 94B05 |
| url | https://arxiv.org/abs/2512.12335 |