Abstract
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 ps, 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 ps 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.
Highlights
Cyclic and negacyclic codes have been studied extensively in many contexts, beginning with their linear versions over finite fields and continuing on to the study of such codes over a finite ring alphabet A
We refer to the ring Rn as the ambient space or ambient ring for the codes
While the literature on cyclic and negacylic codes over chain rings has grown in leaps and bounds, in most instances the studies have been focused only on the cases where the characteristic of the alphabet ring is coprime to the code length, the so-called simple root codes
Summary
Cyclic and negacyclic codes have been studied extensively in many contexts, beginning with their linear versions over finite fields and continuing on to the study of such codes over a finite ring alphabet A. While the literature on cyclic and negacylic codes over chain rings (such as Galois rings) has grown in leaps and bounds (see [4, 10, 11, 17, 18, 21]), in most instances the studies have been focused only on the cases where the characteristic of the alphabet ring is coprime to the code length, the so-called simple root codes. A few of the contributions to the study of the cases where the characteristic of the alphabet ring is not coprime the the code length (repeated root codes) are [1, 2, 5, 9, 16, 19].
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