The Gray image of a class of constacyclic codes over polynomial residue rings

Abstract

Let Rk denote the polynomial residue ring F2m[u]/〈uk〉, where 2j−1+1≤k≤2j for some positive integer j. Motivated by the work in Abular and Siap ([1], 2009) and Qian ([8], 2011), we introduce a new Gray map from Rk to F2m2j. It is proved that the Gray image of a linear (1+u+⋯+uk−1) constacyclic code of an arbitrary length N over Rk is a distance invariant linear cyclic code of length 2jN over F2m. Furthermore, the generator polynomial of the Gray image of such a constacyclic code is determined, and some optimal linear cyclic codes over F2 and F4 are constructed under this Gray map.