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 also given. The Hamming distance of certain constacyclic codes of length ηp s and 2ηp s over Fpm is computed. A method, which determines the Hamming distance of the constacyclic codes of length ηp s and 2ηp 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 2p s over Fpm is determined explicitly.
Dual codes are defined with respect to non-degenerate sesquilinear or bilinear forms over a finite Frobenius ring. These dual codes have the properties one expects from a dual code: they satisfy a double-dual property, they have cardinality complementary to that of the primal code, and they satisfy the MacWilliams identities for the Hamming weight.2010 MSC: 94B05, 15A63
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