GRAY ISOMETRIES FOR FINITE p-GROUPS

We construct two classes of gray maps, called type-i gray map and type-ii gray map,for a finite p-group g. type-i gray maps are constructed based on the existence of a gray map for amaximal subgroup h of g. when g is a semidirect product of two finite p-groups h and k, both hand k admit gray maps and the corresponding homomorphism : h ??! aut(k) is compatible withthe gray map of k in a sense which we will explain, we construct type-ii gray maps for g. finally,we consider group codes over the dihedral group d8 of order 8 given by the set of their generators, andderive a representation and an encoding procedure for such codes.