Enumeration of balanced symmetric functions over GF(p)

Symmetric functions display some interesting properties since this class of functions are invariant under permutation of indices. In this paper, we prove that the construction and enumeration of the number of balanced symmetric functions over GF(p) are equivalent to solving an equation system and enumerating the solutions, as a result we obtain the exact number of n-variable balanced symmetric functions by searching the solutions of the equation system. When n and p become large, we give a lower bound on number of balanced symmetric functions over GF(p), and the lower bound provides best known result.