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Abstract: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 group-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 $\operatorname{char}(\mathbb{F}_{q^2})$, through a suitable Wedderburn-Artin 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 $\operatorname{char}(\mathbb{F}_q)$, via the Wedderburn-Artin 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 an 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, with this methodical approach, we recover some already known quantum codes that achieve the best known minimum distance for their length and dimension.

Submission history

From: Víctor Sotomayor [view email]
[v1] Mon, 8 Dec 2025 09:51:29 UTC (34 KB)
[v2] Wed, 4 Feb 2026 17:14:40 UTC (37 KB)
[v3] Sun, 6 Sep 2026 08:19:33 UTC (38 KB)