p-Adic estimates of Hamming weights in Abelian codes over Galois rings

A generalization of McEliece's theorem on the p-adic valuation of Hamming weights of words in cyclic codes is proved in this paper by means of counting polynomial techniques introduced by Wilson along with a technique known as trace-averaging introduced here. The original theorem of McEliece concerned cyclic codes over prime fields. Delsarte and McEliece later extended this to Abelian codes over finite fields. Calderbank, Li, and Poonen extended McEliece's original theorem to cover cyclic codes over the rings /spl Zopf//sub 2//sup d/, Wilson strengthened their results and extended them to cyclic codes over /spl Zopf//sub p//sup d/, and Katz strengthened Wilson's results and extended them to Abelian codes over /spl Zopf//sub p//sup d/. It is natural to ask whether there is a single analogue of McEliece's theorem which correctly captures the behavior of codes over all finite fields and all rings of integers modulo prime powers. In this paper, this question is answered affirmatively: a single theorem for Abelian codes over Galois rings is presented. This theorem contains all previously mentioned results and more

p-Adic valuation of weights in Abelian codes over /spl Zopf/(p/sup d/)

Counting polynomial techniques introduced by Wilson are used to provide analogs of a theorem of McEliece. McEliece's original theorem relates the greatest power of p dividing the Hamming weights of words in cyclic codes over GF (p) to the length of the smallest unity-product sequence of nonzeroes of the code. Calderbank, Li, and Poonen presented analogs for cyclic codes over /spl Zopf/(2/sup d/) using various weight functions (Hamming, Lee, and Euclidean weight as well as count of occurrences of a particular symbol). Some of these results were strengthened by Wilson, who also considered the alphabet /spl Zopf/(p/sup d/) for p an arbitrary prime. These previous results, new strengthened versions, and generalizations are proved here in a unified and comprehensive fashion for the larger class of Abelian codes over /spl Zopf/(p/sup d/) with p any prime. For Abelian codes over /spl Zopf//sub 4/, combinatorial methods for use with counting polynomials are developed. These show that the analogs of McEliece's theorem obtained by Wilson (for Hamming weight, Lee weight, and symbol counts) and the analog obtained here for Euclidean weight are sharp in the sense that they give the maximum power of 2 that divides the weights of all the codewords whose Fourier transforms have a specified support

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

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. Along with some structure details of the ambient ring, the existance of a certain type of generating set for an ideal is proven.Comment: arXiv admin note: text overlap with arXiv:0906.400