A Note on a Conjecture for Balanced Elementary Symmetric Boolean Functions

In 2008, Cusick {\it et al.} conjectured that certain elementary symmetric Boolean functions of the form ${\sigma }_{2^{t+1}l-1,2^t}$ are the only nonlinear balanced ones, where $t$, $l$ are any positive integers, and ${\sigma }_{n,d}={⨁}_{1\le i_1<\mathrm{.}\mathrm{.}\mathrm{.}<i_d\le n}{x}_{i_1}{x}_{i_2}\mathrm{.}\mathrm{.}\mathrm{.}{x}_{i_d}$ for positive integers $n$, $1\le d\le n$. In this note, by analyzing the weight of ${\sigma }_{n,2^t}$ and ${\sigma }_{n,d}$, we prove that ${\rm wt}(\sigma_{n, d})<2^{n-1}$ holds in most cases, and so does the conjecture. According to the remainder of modulo 4, we also consider the weight of ${\sigma }_{n,d}$ from two aspects: n\equiv 3({\rm mod\}4) and n\not\equiv 3({\rm mod\}4). Thus, we can simplify the conjecture. In particular, our results cover the most known results. In order to fully solve the conjecture, we also consider the weight of ${\sigma }_{n,2^t+2^s}$ and give some experiment results on it.