Paper 2012/101

Unbalanced Elementary Symmetric Boolean Functions with the Degree "d" and "wt(d)>=3"

Zhihui Ou

Abstract

In the paper, for $d=2^{t}k$, $n=2^t(2k+q)+m$ and special $k=2^w(2^0+2^1+\cdots +2^s)$, we present that a majority of $X(d,n)$ are not balanced. The results include many cases $wt(d)\ge 3$ and $n\equiv 0,1,2,3mod4$. The results are also parts of the conjecture that $X(2^t,2^{t+1}l-1)$ is only nonlinear balanced elementary symmetric Boolean function. Where $t\ge 2$, $q\ge 1$, $s\ge 0$, $w\ge 0$ and $m\ge -1$ are integers, and $X(d,n)=\underset{1\le i_1<\cdots <i_d\le n}{⨁}{x}_{i_1}\cdots x_{i_d}$.\newline\newline