Maximum nonlinearity of symmetric Boolean functions on odd number of variables

In this correspondence, we establish that for odd n, the maximum nonlinearity achievable by an n-variable symmetric Boolean function is 2^n-1-2^(n-1)/² and characterize the set of functions which achieve this value of nonlinearity. In particular, we show that for each odd n≥3, there are exactly four possible symmetric Boolean functions achieving the nonlinearity 2^n-1-2^(n-1)2/.