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

Let Fq 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 Fq[G]. In this paper, we provide a complete group-algebraic description for the Hermitian dual code of any Dₙ-code over Fq², where Dₙ is a dihedral group of order $2n$ with n not divisible by char(Fq²), through a suitable Wedderburn-Artin decomposition of the group algebra Fq²[Dₙ], and we determine all distinct Hermitian self-orthogonal Dₙ-codes over Fq². We also present a thorough representation of the Euclidean dual code of any Qₙ-code over Fq, where Qₙ is a generalised quaternion group of order $4n$ not divisible by char(Fq), via the Wedderburn-Artin decomposition of the group algebra Fq[Qₙ]. In particular, since the semisimple group algebras Fq²[Qₙ] and Fq²[D₂ₙ] are isomorphic, then the Hermitian dual code of any Qₙ-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.