Adic Valuation of Weights in
Abelian Codes Over, Daniel J. Katz · 2005
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 di- viding the Hamming weights of words in cyclic codes over GF to the length of the smallest unity-product sequence of nonzeroes of the code. Calderbank, Li, and Poonen presented analogs for cyclic codes over 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 for 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 with any prime. For Abelian codes over , 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 that divides the weights of all the codewords whose Fourier transforms have a specified support.