2025/12/08 by Sales-Cabrera, Miguel, Soler-Escrivà, Xaro, Sotomayor, Víctor
#11T71 #16D25 #20C05 #81P73 #94B05 #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2512.07354
Let \mathbbFq 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 \mathbbFq[G]. In this paper, we provide a complete algebraic description for the hermitian dual code of any Dn-code over \mathbbFq2, where Dn is a dihedral group of order 2n with gcd(q,n)=1, through a suitable Wedderburn-Artin's decomposition of the group algebra \mathbbFq2[Dn], and we determine all distinct hermitian self-orthogonal Dn-codes over \mathbbFq2. We also present a thorough representation of the euclidean dual code of any Qn-code over \mathbbFq, where Qn is a generalised quaternion group of order 4n with gcd(q,4n)=1, via the Wedderburn-Artin's decomposition of the group algebra \mathbbFq[Qn]. In particular, since the semisimple group algebras \mathbbFq2[Qn] and \mathbbFq2[D2n] are isomorphic, then the hermitian dual code of any Qn-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.