Abstract: Based on the torsion codes of a  (1+λu) constacyclic code with arbitrary length over R(pm,k)=Fpm [u]／＜ uk ＞, a bound for the homogeneous distance of a (1+λu) constacyclic code with an arbitrary length over  R（ pm ，k） is obtained and the exact homogeneous distances of some (1+λu) constacyclic codes over  R（ pm ，k） are determined, where λ is a unit of  R（ pm ，k）. Furthermore, a new distance-preserving Gray map from RN（pm ，k） (Homogeneous distance) to  F pm（k －1） N pm (Hamming distance) is defined. It is proved that the Gray image of a linear (1+λu) constacyclic code of arbitrary length over  R（pm ，k） is a linear code over Fpm, and some optimal linear codes over F2, F3, and F4 are constructed under this Gray map.

Key words:  , Optimal code, Constacyclic code, Homogeneous distance, Gray map