In this paper we consider perturbations of symmetric Boolean functions \(\sigma_{n,k_1}+\cdots+\sigma_{n,k_s}\) in
\(n\) variables and degree \(k_s\). We compute the asymptotic behavior of the exponential sum of Boolean
functions of the type
$$\sigma_{n,k_1}+\cdots+\sigma_{n,k_s}+F(X_1,\cdots,X_j)$$
for \(j\) fixed. In particular, we characterize all the Boolean functions of the type
$$\sigma_{n,k_1}+\cdots+\sigma_{n,k_s}+F(X_1,\cdots,X_j)$$
that are asymptotic balanced. We also present an algorithm that computes the asymptotic behavior of a family of
Boolean functions from one member of the family. Finally, as a byproduct of our results, we provide a relation between
the parity of families of sums of binomial coefficients.
