Complete classification of (δ+αu2)(δ+αu2)-constacyclic codes of length pkpk over Fpm+uFpm+u2FpmFpm+uFpm+u2Fpm

Let R   be the finite chain ring Fpm[u]/〈u3〉Fpm[u]/〈u3〉, where p is a prime and m   is a positive integer. In this study we completely determine the structure of (δ+αu2)(δ+αu2)-constacyclic codes of length pkpk over R  , that is, ideals of the ring R[x]/〈xpk−(δ+αu2)〉R[x]/〈xpk−(δ+αu2)〉, where δ and α   are nonzero elements in FpmFpm. We show that when p   is odd, there is no self-dual (δ+αu2)(δ+αu2)-constacyclic code of length pkpk over R   and also in the case where p=2p=2, self-dual codes exist when δ=1δ=1. We completely determine self-dual (1+αu2)(1+αu2)-constacyclic codes of length 2k2k over F2m[u]/〈u3〉F2m[u]/〈u3〉 and enumerate them.