Dualities of dihedral and generalised quaternion codes and applications to quantum codes
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2025
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| author | Sales-Cabrera, Miguel Soler-Escrivà, Xaro Sotomayor, Víctor |
| author_facet | Sales-Cabrera, Miguel Soler-Escrivà, Xaro Sotomayor, Víctor |
| contents | Let $\mathbb{F}_q$ be a finite field of $q$ elements, for some prime power $q$, and let $G$ be a finite group. A (left) group code, or simply a $G$-code, is a (left) ideal of the group algebra $\mathbb{F}_q[G]$. In this paper, we provide a complete algebraic description for the hermitian dual code of any $D_n$-code over $\mathbb{F}_{q^2}$, where $D_n$ is a dihedral group of order $2n$ with $n$ not divisible by char$(\mathbb{F}_{q^2})$, through a suitable Wedderburn-Artin's decomposition of the group algebra $\mathbb{F}_{q^2}[D_n]$, and we determine all distinct hermitian self-orthogonal $D_n$-codes over $\mathbb{F}_{q^2}$. We also present a thorough representation of the euclidean dual code of any $Q_n$-code over $\mathbb{F}_q$, where $Q_n$ is a generalised quaternion group of order $4n$ not divisible by char$(\mathbb{F}_q)$, via the Wedderburn-Artin's decomposition of the group algebra $\mathbb{F}_q[Q_n]$. In particular, since the semisimple group algebras $\mathbb{F}_{q^2}[Q_n]$ and $\mathbb{F}_{q^2}[D_{2n}]$ are isomorphic, then the hermitian dual code of any $Q_n$-code has also been fully described. As application of the hermitian dualities computed, we give a systematic construction, via the structure of the group algebra, to obtain quantum error-correcting codes, and in fact we rebuild some already known optimal quantum codes with this methodical approach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_07354 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dualities of dihedral and generalised quaternion codes and applications to quantum codes Sales-Cabrera, Miguel Soler-Escrivà, Xaro Sotomayor, Víctor Information Theory Quantum Algebra Rings and Algebras 94B05, 20C05, 11T71, 16D25, 81P73 Let $\mathbb{F}_q$ be a finite field of $q$ elements, for some prime power $q$, and let $G$ be a finite group. A (left) group code, or simply a $G$-code, is a (left) ideal of the group algebra $\mathbb{F}_q[G]$. In this paper, we provide a complete algebraic description for the hermitian dual code of any $D_n$-code over $\mathbb{F}_{q^2}$, where $D_n$ is a dihedral group of order $2n$ with $n$ not divisible by char$(\mathbb{F}_{q^2})$, through a suitable Wedderburn-Artin's decomposition of the group algebra $\mathbb{F}_{q^2}[D_n]$, and we determine all distinct hermitian self-orthogonal $D_n$-codes over $\mathbb{F}_{q^2}$. We also present a thorough representation of the euclidean dual code of any $Q_n$-code over $\mathbb{F}_q$, where $Q_n$ is a generalised quaternion group of order $4n$ not divisible by char$(\mathbb{F}_q)$, via the Wedderburn-Artin's decomposition of the group algebra $\mathbb{F}_q[Q_n]$. In particular, since the semisimple group algebras $\mathbb{F}_{q^2}[Q_n]$ and $\mathbb{F}_{q^2}[D_{2n}]$ are isomorphic, then the hermitian dual code of any $Q_n$-code has also been fully described. As application of the hermitian dualities computed, we give a systematic construction, via the structure of the group algebra, to obtain quantum error-correcting codes, and in fact we rebuild some already known optimal quantum codes with this methodical approach. |
| title | Dualities of dihedral and generalised quaternion codes and applications to quantum codes |
| topic | Information Theory Quantum Algebra Rings and Algebras 94B05, 20C05, 11T71, 16D25, 81P73 |
| url | https://arxiv.org/abs/2512.07354 |