Communication complexities of symmetric XOR functions

We call F : {0, 1}n×{0, 1}n → {0, 1} a symmetric XOR function if for a function S : {0, 1, ..., n} → {0, 1}, F (x, y) = S(|x⊕ y|), for any x, y ∈ {0, 1}n, where |x⊕ y| is the Hamming weight of the bit-wise XOR of x and y. We show that for any such function, (a) the deterministic communication complexity is always Θ(n) except for four simple functions that have a constant complexity, and (b) up to a polylog factor, the error-bounded randomized and quantum communication complexities are Θ(r0 + r1), where r0 and r1 are the minimum integers such that r0, r1 ≤ n/2 and S(k) = S(k + 2) for all k ∈ [r0, n− r1).