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
@article{1175789804,
author = {Cusick, Thomas W. and Cheon, Younhwan},
title = {Counting Balanced Boolean Functions in $n$ Variables with Bounded Degree},
journal = {Experiment. Math.},
volume = {16},
number = {1},
year = {2007},
pages = { 101-106},
language = {en},
url = {http://dml.mathdoc.fr/item/1175789804}
}
Cusick, Thomas W.; Cheon, Younhwan. Counting Balanced Boolean Functions in $n$ Variables with Bounded Degree. Experiment. Math., Tome 16 (2007) no. 1, pp. 101-106. http://gdmltest.u-ga.fr/item/1175789804/