Dualities of dihedral and generalised quaternion codes and applications to quantum codes

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.