Quantum Codes from Group Codes

Abstract

We study linear codes and quantum error-correcting codes (QECCs) constructed from group rings over finite fields. Using the algebraic structure of group rings, we give a single framework for codes over several group structures, including cyclic, dihedral, direct-product, and semidirect-product groups. We establish necessary and sufficient conditions for these group codes to be self-orthogonal under the Euclidean, Hermitian, and symplectic inner products. We show that non-isomorphic groups of the same order can generate inequivalent codes with distinct parameters, and we support this with explicit computational comparisons. Using these structural results, we give explicit constructions of quantum codes and provide new examples that match or improve upon the best known parameters. In particular, we describe explicit block-matrix forms of the generating matrices for dihedral and direct-product groups, and we use a Kronecker-product construction to obtain an infinite family of self-orthogonal group codes together with the corresponding QECCs.