The trace dual of nonlinear skew cyclic codes
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915964815147008 |
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| author | Bossaller, Daniel Herden, Daniel Ruiz-Bolaños, Indalecio |
| author_facet | Bossaller, Daniel Herden, Daniel Ruiz-Bolaños, Indalecio |
| contents | Codes which have a finite field $\mathbb{F}_{q^m}$ as their alphabet but which are only linear over a subfield $\mathbb{F}_q$ are a topic of much recent interest due to their utility in constructing quantum error correcting codes. In this article, we find generators for trace dual spaces of different families of $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^2}$. In particular, given the field extension $\mathbb{F}_q\leq \mathbb{F}_{q^2}$ with $q$ an odd prime power, we determine the trace Euclidean and trace Hermitian dual codes for the general $\mathbb{F}_q$-linear cyclic $\mathbb{F}_{q^2}$-code. In addition, we also determine the trace Euclidean and trace Hermitian duals for general $\mathbb{F}_q$-linear skew cyclic $\mathbb{F}_{q^2}$-codes, which are defined to be left $\mathbb{F}_q[X]$-submodules of $\mathbb{F}_{q^2}[X;σ]/(X^n-1)$, where $σ$ denotes the Frobenius automorphism and $\mathbb{F}_{q^2}[X;σ]$ the induced skew polynomial ring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09098 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The trace dual of nonlinear skew cyclic codes Bossaller, Daniel Herden, Daniel Ruiz-Bolaños, Indalecio Information Theory Rings and Algebras 94B05, 94B15, 11T71, 16S36 Codes which have a finite field $\mathbb{F}_{q^m}$ as their alphabet but which are only linear over a subfield $\mathbb{F}_q$ are a topic of much recent interest due to their utility in constructing quantum error correcting codes. In this article, we find generators for trace dual spaces of different families of $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^2}$. In particular, given the field extension $\mathbb{F}_q\leq \mathbb{F}_{q^2}$ with $q$ an odd prime power, we determine the trace Euclidean and trace Hermitian dual codes for the general $\mathbb{F}_q$-linear cyclic $\mathbb{F}_{q^2}$-code. In addition, we also determine the trace Euclidean and trace Hermitian duals for general $\mathbb{F}_q$-linear skew cyclic $\mathbb{F}_{q^2}$-codes, which are defined to be left $\mathbb{F}_q[X]$-submodules of $\mathbb{F}_{q^2}[X;σ]/(X^n-1)$, where $σ$ denotes the Frobenius automorphism and $\mathbb{F}_{q^2}[X;σ]$ the induced skew polynomial ring. |
| title | The trace dual of nonlinear skew cyclic codes |
| topic | Information Theory Rings and Algebras 94B05, 94B15, 11T71, 16S36 |
| url | https://arxiv.org/abs/2504.09098 |