On p-Adic Estimates of Weights in Abelian Codes over Galois Rings

Ax-Katz;Chevalley-Warning;cyclic codes;Delsarte;error-correcting codes;McEliece;p-divisibility;polynomials

Katz, Daniel Jerome ; Wilson, Richard M.

University:California Institute of Technology

Department:Physics, Mathematics and Astronomy

关键词: Ax-Katz;    Chevalley-Warning;    cyclic codes;    Delsarte;    error-correcting codes;    McEliece;    p-divisibility;    polynomials;   

Others  :  https://thesis.library.caltech.edu/2329/1/djkatz_thesis.pdf

美国|英语

来源: Caltech THESIS

【 摘 要 】

Let p be a prime.We prove various analogues and generalizations of McEliece's theorem on the p-divisibility of weights of words in cyclic codes over a finite field of characteristic p.Here we consider Abelian codes over various Galois rings.We present four new theorems on p-adic valuations of weights.For simplicity of presentation here, we assume that our codes do not contain constant words.

The first result has two parts, both concerning Abelian codes over Z/p^dZ.The first part gives a lower bound on the p-adic valuations of Hamming weights.This bound is shown to be sharp: for each code, we find the maximum k such that p^k divides all Hamming weights.The second part of our result concerns the number of occurrences of a given nonzero symbol s ∈ Z/p^dZ in words of our code; we call this number the s-count.We find a j such that p^j divides the s-counts of all words in the code.Both our bounds are stronger than previous ones for infinitely many codes.

The second result concerns Abelian codes over Z/4Z.We give a sharp lower bound on the 2-adic valuations of Lee weights.It improves previous bounds for infinitely many codes.

The third result concerns Abelian codes over arbitrary Galois rings.We give a lower bound on the p-adic valuations of Hamming weights.When we specialize this result to finite fields, we recover the theorem of Delsarte and McEliece on the p-divisibility of weights in Abelian codes over finite fields.

The fourth result generalizes the Delsarte-McEliece theorem.We consider the number of components in which a collection c_1,...,c_t of words all have the zero symbol; we call this the simultaneous zero count.Our generalized theorem p-adically estimates simultaneous zero counts in Abelian codes over finite fields, and we can use it to prove the theorem of N. M. Katz on the p-divisibility of the cardinalities of affine algebraic sets over finite fields.