New families of balanced symmetric functions and a generalization of Cusick, Li and Stǎnicǎ’s conjecture

Value distribution of elementary symmetric polynomials and its perturbations over finite fields

“…k (t) with equality only for k = d · p l where 1 ≤ d ≤ p − 1. Fine also proved (3) for all p, which implies that the proof of the generalization of the conjecture of Cusick, Li and Stǎnicǎ presented in [2] is expected to be much harder than the binary counterpart. In particular, when p > 2, the approach presented in [6] will fail to prove the conjecture asymptotically when k = d · p l and 1 < d ≤ p − 1.…”

“…A generalized version of this conjecture for finite fields was presented in [2]. It is known that Conjecture 2.1 is true asymptotically [6,15,16].…”

“…In [26], J. D. Smith generalized Aberth's example and showed that if p > 3 is prime, then As mentioned at the beginning of Section 2, one of the reasons the asymptotic behavior of exponential sums of symmetric polynomials was calculated over the binary field was to provide an asymptotic proof of Conjecture 2.1 (see [6]). The concept of asymptotically balanced symmetric Boolean function was introduced in [6] to mean that c (2) A (k; −1) = 0. Conjecture 2.1 was proved asymptotically in [6] by showing that e n,k is asymptotically balanced if and only if k is a power of two.…”

“…Observe that Fine [15] proved that e n,k is asymptotically balanced over the prime field F p when k = d · p l where 1 ≤ d ≤ p − 1. In [2,Th. 2], it was proved that if q = p r , then e n,p is asymptotically balanced over F q for every .…”

“…[2]). The only nonlinear balanced elementary symmetric polynomial over F q , q = p r are those with degree k = p and n = p D − 1 variables, where , D ∈ N, D ≡ 1 mod p.…”