  We consider the simplified Algebraic Normal Form (sANF) of Boolean functions
vanishing on Hamming spheres centred at zero and the associated sANF vector. We
show that this vector is periodic, leading to an efficient computation of the
sANF and to specific formulas for particular cases. Moreover, we explicitly
provide a connection to the binary M{\"o}bius transform of the elementary
symmetric functions. We conclude by presenting a method based on polynomial
evaluation to bound the minimum distance of binary nonlinear codes. The same
method can be used to compute the minimum distance and the weight distribution
of binary linear codes.
