The Degree of Balanced Elementary Symmetric Boolean Functions of ${{\bf 4k}+{\bf 3}}$ Variables

In this paper, we consider the conjecture that σ₂t+1l-1,2t are the only nonlinear balanced elementary symmetric Boolean functions where t and l are positive integers. We prove if n=2t+1l-1, l odd and 2t+1nmid d, σn,d is balanced if and only if d=2k, 1 ≤ k ≤ t. Our results verify most cases of the conjecture for n ≡ 3 (mod 4) .