Codes in Dihedral Group Algebra

Abstract

Robert McEliece developed an asymmetric encryption algorithm based on the use of binary Goppa codes in 1978 and no effective key attacks has been described yet. Variants of this cryptosystem are known due to the use of different codes types, but most of them were proven to be less secure. Code cryptosystems are considered an alternate to number-theoretical ones in connection with the development of quantum computing. So, the new classes of error-correcting codes are required for building new resistant code cryptosystems. Non-commutative codes, which simply are ideals of finite non-commutative group algebras, are an option. The Artin–Wedderburn theorem implies that a group algebra is isomorphic to a finite direct sum of matrix algebras, when the order of the group and the field characteristics are relatively prime. This theorem is important to study the structure of a non-commutative code, but it gives no information about summands and the isomorphism. In case of a dihedral group these summands and the isomorphism were found by F. E. Brochero Martinez. The purpose of the paper is to study codes in dihedral group algebras as and when the order of a group and a field characteristics are relatively prime. Using the result of F. E. Brochero Martinez, we consider a structure of all dihedral codes in this case and the codes induced by cyclic subgroup codes.