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 $\sigma{n,d}=\bigoplus{1\le i1<...<id\le n}x{i1}x{i2}...x{id}$ 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 $\sigma{n, 2^t+2s}$ and give some experiment results on it.