On theorems of Delsarte–McEliece and Chevalley–Warning–Ax–Katz

Abstract

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 $${(c^{(1)}_i,\ldots,c^{(t)}_i)\not=(0,\ldots,0)}$$ . 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 $${(c^{(1)}_i,\ldots,c^{(t)}_i)=(s_1,\ldots,s_t)}$$ as these words range over a product of abelian codes.