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 ( p a , 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 p s over GR ( p 2 , m ) . The Hamming distance of certain constacyclic codes of length η p s over F p m is computed. A method, which determines the Hamming distance of the constacyclic codes of length η p s over GR ( p a , m ) , where ( η , p ) = 1 , is described. In particular, the Hamming distance of all cyclic codes of length p s over GR ( p 2 , m ) and all negacyclic codes of length 2 p s over F p m is determined explicitly.