A Note on a Conjecture for Balanced Elementary Symmetric Boolean Functions
Wei Yi Su, Xiaohu Tang, Alexander Pott · IEEE Transactions on Information Theory · 2012
In 2008, Cusick et al conjectured that certain elementary symmetric Boolean functions of the form σ(2t+1)l-1, (2t) are the only nonlinear balanced ones, wheret,lare any positive integers, and σn,d=⊕1≤( i1)d≤ n xi1xi2⋯xid) for positive integersn, 1 ≤d≤n. In this paper, by analyzing the weight of σn,(2t) and σn, d, we prove that wt σn,dn-1holds in most cases, and so does the conjecture. According to the remainder modulo 4, we also consider the weight of σn, dfrom two aspects: n ≠ 3(mod 4) and n not ≡ 3(mod 4). In particular, our results not only cover the most known results, but also contain some new cases. Thus, we can reduce the conjecture to few remaining cases. We do not fully solve the conjecture, but we also consider the weight of σn, (2t+2s) and also give some experimental results on it.