Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes

Abstract

Cyclic, negacyclic and constacyclic codes are part of a larger class of codes called polycyclic codes; namely, those codes which can be viewed as ideals of a factor ring of a polynomial ring. The structure of the ambient ring of polycyclic codes over GR(pa,m) and generating sets for its ideals are considered. It is shown that these generating sets are strong Groebner bases. A method for finding such sets in the case that a=2 is given. This explicitly gives the Hamming distance of all cyclic codes of length ps over GR(p2,m). The Hamming distance of certain constacyclic codes of length ηps over Fpm is computed. A method, which determines the Hamming distance of the constacyclic codes of length ηps over GR(pa,m), where (η,p)=1, is described. In particular, the Hamming distance of all cyclic codes of length ps over GR(p2,m) and all negacyclic codes of length 2ps over Fpm is determined explicitly.