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 $wt(d)\geq 3$ and $n\equiv 0,1,2,3 mod4$. 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 $t\geq 2$, $q\geq 1$, $s\geq 0$, $w\geq 0$ and $m\geq -1$ are integers, and $X(d,n)=\bigoplus\limits_{1\le i_{1} <\cdots<i_{d}\le n} x_{i_{1} } \cdots x_{i_{d} }$.\newline\newline