Improved lower bound on the number of balanced symmetric functions over GF(p)

The lower bound on the number of n  -variable balanced symmetric functions over finite fields GF(p)GF(p) presented by Cusick et al. in [T.W. Cusick, Y. Li, P. Staˇnicaˇ, Balanced symmetric functions over GF(p), IEEE Trans. Inform. Theory 54 (3) (2008) 1304–1307] is improved in this paper. An equivalent characterization is also presented for the general case.