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 a maximal‎ ‎subgroup H of G‎. ‎When G is a semidirect product of two‎ ‎finite p-groups H and K‎, ‎both H and K admit Gray maps‎ ‎and the corresponding homomorphism \psi:H\longrightarrow {\rm‎ ‎Aut}(K) is compatible with the 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 D_۸ of order ۸‎ ‎given by the set of their generators‎, ‎and derive a representation‎ ‎and an encoding procedure for such codes‎.

Keywords:

Finite group , Code , Gray map , Isometry