On Repeated-Root Constacyclic Codes of Prime Power Length Over Polynomial Residue Rings

The polynomial residue ring R-a = F(p)m[u]/< u(a)> = F(p)m + uF(p)n + ... + u(a-1)F(p)m is a chain ring with residue field F(p)m, that contains precisely (p(m)-1)p(m(a-1)) units, namely, alpha(0) + u alpha(1) + ... + u(a-1) alpha(a-1) where alpha(0), alpha(1), ... , alpha(a-1) is an element of F(p)m, alpha(0) not equal 0. We classify these units into a-1 types, and show that any constacyclic code of length p(s) of the type k is in a one-to-one correspondence to a constacyclic code of length p(s) of simpler type k* via a ring isomorphism. Two classes of units of R-a, are considered in details, namely, lambda = 1 +u lambda 1 + ... +u(a-l)lambda(a-1), where lambda(1), ... , lambda(a-1) is an element of F(p)m, lambda(1) not equal 0; and Lambda = Lambda(0) +u Lambda(1) + ... +u(a-1)Lambda(a-1), where Lambda(0), Lambda(1), ... , Lambda(a-1) is an element of F(p)m, Lambda(0) not equal 0, Lambda(1) not equal 0. Among other results, the structure, Hamming and homogeneous distances of lambda- and Lambda-constacyclic codes of length p(s) over R-a are established.