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 $\gcd(q,n)=1$, 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 $4n$ with $\gcd(q,4n)=1$, 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.

翻译：设 $\mathbb{F}_q$ 为具有 $q$ 个元素的有限域，其中 $q$ 为某个素数的幂，并设 $G$ 为有限群。一个（左）群码，或简称为 $G$-码，是群代数 $\mathbb{F}_q[G]$ 的一个（左）理想。本文通过群代数 $\mathbb{F}_{q^2}[D_n]$ 的适当 Wedderburn-Artin 分解，完整地描述了任意 $\mathbb{F}_{q^2}$ 上 $D_n$-码的 Hermitian 对偶码的代数结构，其中 $D_n$ 是阶为 $2n$ 的二面体群且满足 $\gcd(q,n)=1$，并确定了所有不同的 Hermitian 自正交 $D_n$-码。我们还通过群代数 $\mathbb{F}_q[Q_n]$ 的 Wedderburn-Artin 分解，详细刻画了任意 $\mathbb{F}_q$ 上 $Q_n$-码的欧几里得对偶码，其中 $Q_n$ 是阶为 $4n$ 的广义四元数群且满足 $\gcd(q,4n)=1$。特别地，由于半单群代数 $\mathbb{F}_{q^2}[Q_n]$ 与 $\mathbb{F}_{q^2}[D_{2n}]$ 同构，因此任意 $Q_n$-码的 Hermitian 对偶码也得以完整描述。作为所计算的 Hermitian 对偶性的应用，我们通过群代数的结构给出了一种系统构造量子纠错码的方法，并利用这一系统性方法重建了一些已知的最优量子码。