On theorems of Delsarte-McEliece and Chevalley-Warning-Ax-Katz
We present a theorem that generalizes the result of Delsarte and McEliece on the p-divisibilities of weights in abelian codes. Our result generalizes the Delsarte-McEliece theorem in the same sense that the theorem of N. M. Katz generalizes the theorem of Ax on the p-divisibilities of cardinalities of affine algebraic sets over finite fields. As the Delsarte-McEliece theorem implies the theorem of Ax, so our generalization implies that of N. M. Katz. The generalized theorem gives the p-divisibility of the t-wise Hamming weights of t-tuples of codewords (c((1)), ... , c((t))) as these words range over a product of abelian codes, where the t-wise Hamming weight is defined as the number of positions i in which the codewords do not simultaneously vanish, i. e., for which (c(i)((1)), ... , c(i)((t))) not equal (0, ... , 0). We also present a version of the theorem that, for any list of t symbols s(1), ... , s(t), gives p-adic estimates of the number of positions i such that (c(i)((1)), ... , c(i)((t))) = (s(1), ... , s(t)) as these words range over a product of abelian codes.
delsarte-mceliece abelian codes algebraic sets cyclic codes ax-katz