Gray Images of Constacyclic Codes over some Polynomial Residue Rings

Let  be the quotient ring    where  is the finite field of size   and  is a positive integer. A Gray map  of length  over  is a special map from  to ( . The Gray map   is said to be a ( )-Gray map if the image of any -constacyclic code over    is a -constacyclic code over the field   . In this paper we investigate the existence of   ( )-Gray maps over . In this direction, we find an equivalent condition for such maps. Then we prove that if  is a        ( )-Gray map of length  over  and also  then we must have       ,   and  divides the length of  that is . We also prove that if   then we must have      . Moreover, we determine all of such maps over . Finally, we introduce a             ( , )-Gray map of length    over  where .