Counting Balanced Boolean Functions in $n$ Variables with Bounded Degree

Abstract

We consider the problem of obtaining good upper and lower bounds on the number of balanced Boolean functions in $n$ variables with degree less than or equal to $k$. This is the same as the problem of finding bounds on the number of codewords of weight $2^{n-1}$ in the Reed--Muller code of length $2^n$ and order $k$. We state several conjectures and use them to obtain good bounds. We believe that the conjectures will be highly useful for further research