In this paper we obtain the Wedderburn-Artin decomposition of a semisimple group algebra associated to a direct product of finite groups. We also provide formulae for the number of all possible group codes, and their dimensions, that can be constructed in a group algebra. As particular cases, we present the complete algebraic description of the group algebra of any direct product of groups whose direct factors are cyclic, dihedral, or generalised quaternion groups. Finally, in the specific case of semisimple dihedral group algebras, we give a method to build quantum error-correcting codes, based on the CSS construction.

翻译：本文中，我们获得了与有限群直积相关联的半单群代数的Wedderburn-Artin分解。我们还给出了在群代数中可构造的所有可能群码的数量及其维度的计算公式。作为特例，我们给出了直因子为循环群、二面体群或广义四元数群的任意群直积的群代数的完整代数描述。最后，针对半单二面体群代数的具体情况，我们提出了一种基于CSS构造构建量子纠错码的方法。