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

Abstract

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/pdZ. 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 pk divides all Hamming weights. The second part of our result concerns the number of occurrences of a given nonzero symbol s ∈ Z/pdZ in words of our code; we call this number the s-count. We find a j such that pj 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.