  Repeated root Cyclic and Negacyclic codes over Galois rings have been studied
much less than their simple root counterparts. This situation is beginning to
change. For example, repeated root codes of length $p^s$, where $p$ is the
characteristic of the alphabet ring, have been studied under some additional
hypotheses. In each one of those cases, the ambient space for the codes has
turned out to be a chain ring. In this paper, all remaining cases of cyclic and
negacyclic codes of length $p^s$ over a Galois ring alphabet are considered. In
these cases the ambient space is a local ring with simple socle but not a chain
ring. Nonetheless, by reducing the problem to one dealing with uniserial
subambients, a method for computing the Hamming distance of these codes is
provided.
