The Log-Rank Conjecture for Read- k XOR Functions

The log-rank conjecture states that the deterministic communication complexity of a Boolean function g (denoted by D(g)) is polynomially related to the logarithm of the rank of the communication matrixMg whereMg is the communication matrix defined byMg(x, y) = g(x, y). An XOR function F : {0, 1} × {0, 1} → {0, 1} with respect to f : {0, 1} → {0, 1} is a function defined by F (x, y) = f(x⊕ y). It is well-known that ||f̂ ||0 = rank(MF ) where ||f̂ ||0 is the Fourier 0-norm of f , MF is the communication matrix defined by MF (x, y) = F (x, y), and rank(MF ) is the dimension of the row space of MF over reals. The log-rank conjecture for XOR functions is equivalent to the question whether the deterministic communication complexity of an XOR function F with respect to a function f is polynomially related to the logarithm of the Fourier sparsity of f , namely D(F ) = log(||f̂ ||0) for a fixed constant c. Previously, the logrank conjecture holds for XOR functions with respect to symmetric functions, linear threshold functions, monotone functions, AC functions, and constant-degree polynomials over F2. In this paper, we consider a special class of functions called read-k polynomials over F2. We study the communication complexity of the XOR function F with respect to a read-k polynomial f . We show that D(F ) = O(kd log(||f̂ ||1)) where d is the F2-degree of f . By the well-known bound that d ≤ log(||f̂ ||0), we conclude that D(F ) = O(k log(||f̂ ||0)). In particular, if k ≤ log(||f̂ ||0), then we have D(F ) = O(log(rank(MF ))).