Semisimple group codes and dihedral codes

Abstract

We consider codes that are given as two-sided ideals in a semisimple finite group algebra \({\mathbb F}_q G\)  defined by idempotents constructed from subgroups of \(G\) in a natural way and compute their dimensions and weights. We give a criterion to decide when these ideals are all the minimal two-sided ideals of \({\mathbb F}_q G\) in the case  when \(G\) is a dihedral group and extend these results also to a family of quaternion group codes. In the final section, we give a method of decoding; i.e., of finding and correcting eventual transmission errors.