On theorems of Delsarte-McEliece and Chevalley-Warning-Ax-Katz (Q690673)

Many results in coding theory about individual words drawn from a code have generalizations that consider multiple codewords. Abelian codes over a finite field are those that come from ideals in certain group algebras. There is a connection between Hamming weights of elements of abelian codes and cardinalities of algebraic sets over \(F_q^n\). For example, a theorem of \textit{P. Delsarte} and \textit{R. J. McEliece} [Am. J. Math. 98, 197--224 (1976; Zbl 0346.12011)] on the \(p\)-divisbilities of Hamming weights of words of abelian codes implies a theorem of \textit{J. Ax} [Am. J. Math. 86, 255--261 (1964; Zbl 0121.02003)] on the \(p\)-divisibilities of sizes of zero sets of polynomials over finite fields. There is a generalization of Ax's result due to \textit{N. M. Katz} [Am. J. Math 93, 485--499 (1971; Zbl 0237.12012)] that concerns the \(p\)-divisibility of the cardinality of the intersection of several polynomials. The goal of this paper is to prove a generalization of the Delsarte-McEliece result that concerns \(p\)-divisibilities of the \(t\)-wise Hamming weights of a \(t\)-tuple of words drawn from a product of abelian codes. This theorem implies the result of N. M. Katz. This paper is written in the language of Fourier analysis of group algebras over finite fields. A codeword is identified with its Fourier coefficients and is then transferred to the \(p\)-adic setting by means of the Teichmüller lift, giving a \(p\)-adic representative for each element of a certain finite field. A key idea of the proofs in this paper is to consider the zero count polynomial of a product of abelian codes. This is a polynomial with coefficients in \(Z_p\) that keeps track of the number of coordinates on which \(t\) codewords simultaneously vanish. Restrictions are then given on the monomials that occur in the zero count polynomial and a lemma of \textit{R. M. Wilson} [Discrete Math. 306, No. 23, 3154--3165 (2006; Zbl 1156.11009)] is used to give estimates for the \(p\)-adic valuations of its coefficients.