When isometry and equivalence for skew constacyclic codes coincide
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910123821105152 |
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| author | Nevins, Monica Pumpluen, Susanne |
| author_facet | Nevins, Monica Pumpluen, Susanne |
| contents | We work in the setting of linear skew constacyclic codes over a commutative base ring $S$. We show that the notions of $(n,σ)$-isometry and $(n,σ)$-equivalence introduced by Ou-azzou et al coincide for most skew $(σ,a)$-constacyclic codes of length $n$. To prove this, we show that all Hamming-weight preserving isomorphisms between their ambient rings which extend some automorphism $τ$ of $S$ that commutes with $σ$ must have degree one, when those rings are not associative. In the process we determine isomorphisms between their nonassociative ambient rings, the Petit rings $S[t;σ]/S[t;σ](t^n-a)$, which give rise to skew constacyclic codes. As a consequence, we propose new definitions of equivalence and isometry of skew constacyclic codes that exactly capture all Hamming-weight preserving isomorphisms between the ambient rings of skew constacyclic codes which extend $τ\in {\rm Aut}(S)$ that commute with $σ$, and lead to tighter classifications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06695 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | When isometry and equivalence for skew constacyclic codes coincide Nevins, Monica Pumpluen, Susanne Information Theory Rings and Algebras 94B15, 11T71, 17A99, 81P70 We work in the setting of linear skew constacyclic codes over a commutative base ring $S$. We show that the notions of $(n,σ)$-isometry and $(n,σ)$-equivalence introduced by Ou-azzou et al coincide for most skew $(σ,a)$-constacyclic codes of length $n$. To prove this, we show that all Hamming-weight preserving isomorphisms between their ambient rings which extend some automorphism $τ$ of $S$ that commutes with $σ$ must have degree one, when those rings are not associative. In the process we determine isomorphisms between their nonassociative ambient rings, the Petit rings $S[t;σ]/S[t;σ](t^n-a)$, which give rise to skew constacyclic codes. As a consequence, we propose new definitions of equivalence and isometry of skew constacyclic codes that exactly capture all Hamming-weight preserving isomorphisms between the ambient rings of skew constacyclic codes which extend $τ\in {\rm Aut}(S)$ that commute with $σ$, and lead to tighter classifications. |
| title | When isometry and equivalence for skew constacyclic codes coincide |
| topic | Information Theory Rings and Algebras 94B15, 11T71, 17A99, 81P70 |
| url | https://arxiv.org/abs/2508.06695 |