Constacyclic codes of arbitrary lengths over ring $$Z_{p^m } + vZ_{p^m }$$

By constructing a Gray map, constacyclic codes of arbitrary lengths over ring $$R = Z_{p^m } + vZ_{p^m }$$ are studied, where v2 = v The structure of constacyclic codes over R and their dual codes are obtained. A necessary and sufficient condition for a linear code to be self-dual constacyclic is given. In particular, (1+(v+1)αp)-constacyclic codes over R are classified in terms of generator polynomial, where α is a unit of $$Z_{p^m }$$.