Designs, Codes and Cryptography

, Volume 65, Issue 3, pp 291–324 | Cite as

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

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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 s1, . . . ,st, 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.

Keywords

Cyclic codes Abelian codes Algebraic sets Delsarte–McEliece Ax–Katz 

Mathematics Subject Classification

94B15 11T71 11T06 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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