The log-rank conjecture states that the deterministic communication complexity of a Boolean function *g* (denoted by *D*^{cc}(*g*)) is polynomially related to the logarithm of the rank of the communication matrix *M*_{g} where *M*_{g} is the communication matrix defined by *M*_{g}(*x*, *y*)=*g*(*x*, *y*). An XOR function *G*:{0, 1}^{n}x{0, 1}^{n}→{0, 1} with respect to *g*:{0, 1}^{n}→{0, 1} is a function defined by *G*(*x*, *y*)= *g*(*x*+*y*). It is well-known that ||*ĝ*||_{0} = **rank**(*M*_{G}) where ||*ĝ*||_{0} is the Fourier 0-norm of *g*, *M*_{G} is the communication matrix defined by *M*_{G}(*x*, *y*)=*G*(*x*, *y*), and **rank**(*M*_{G}) is the dimension of the row space of *M*_{G} over reals. The log-rank conjecture for XOR functions is equivalent to the question whether the deterministic communication complexity of an XOR function *G* with respect to a function *G* is polynomially related to the logarithm of the Fourier sparsity of *g*, namely *D*^{cc}(*G*)=log^{c} (||*ĝ*||_{0}) for a fixed constant *c*. Previously, the log-rank conjecture holds for XOR functions with respect to symmetric functions, linear threshold functions, monotone functions, AC^{0} functions, and constant-degree polynomials over F_{2}.

In this paper, we consider a special class of functions called read-*k* polynomials over F_{2}. We study the communication complexity of the XOR function *G* with respect to a read-*k* polynomial *g*. We show that *D*^{cc}(*G*)=*O*(*kd*^{2}log(||*ĝ*||_{1})) where *d* is the F_{2}-degree of *g*. By the well-known bound that* d* <= log(||*ĝ*||_{0}), we conclude that *D*^{cc}(*G*) = *O*(*k*log^{3}(||*ĝ*|||_{0})). In particular, if *k*=log^{O(1)}(||*ĝ*|||_{0}), then we have *D*^{cc}(*G*)=*O*(log^{O(1)}(**rank**(*M*_{G}))).