Balanced Symmetric Functions over GF(p)

2006/08/15 by Thomas W. Cusick, Yuan Li, Cusick, Thomas W. +4
Computer Science · Mathematics · #05A10 #05A16 #11T71 #Advanced Algebra and Geometry #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #math.CO #msc:05A10 #msc:05A16 #msc:11T71

paper · pdf · doi:10.48550/arxiv.math/0608369

21 pages

arxiv created 2006/08/15 · openalex publication_date 2006/08/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Under mild conditions on n,p, we give a lower bound on the number of n-variable balanced symmetric polynomials over finite fields GF(p), where p is a prime number. The existence of nonlinear balanced symmetric polynomials is an immediate corollary of this bound. Furthermore, we conjecture that X(2^t,2^t+1l-1) are the only nonlinear balanced elementary symmetric polynomials over GF(2), where X(d,n)=∑_i1<i2<...<idx_i1 x_i2... x_id, and we prove various results in support of this conjecture.

Citations

• Maximum nonlinearity of symmetric Boolean functions on odd number of variables

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