Mean and variance of the number of subfunctions of random Boolean function which are close to the affine functions set

Abstract

Abstract For random equiprobable Boolean functions we investigate the distribution of the number of subfunctions which have a given number of variables and are close to the set of affine Boolean functions. It is shown, for example, that for Boolean functions of n variables the mean number of subfunctions having s ⩾ 3 + log2 n variables and the Hamming distance to the set of affine functions smaller than 2 s−2 tends to 0 as n → ∞.