Min-rank conjecture for log-depth circuits

A completion of an m-by-n matrix A with entries in {0,1,@?} is obtained by setting all @?-entries to constants 0 and 1. A system of semi-linear equations over GF"2 has the form Mx=f(x), where M is a completion of A and f:{0,1}^n-{0,1}^m is an operator, the ith coordinate of which can only depend on variables corresponding to @?-entries in the ith row of A. We conjecture that no such system can have more than 2^n^-^@e^@?^m^r^(^A^) solutions, where @e0 is an absolute constant and mr(A) is the smallest rank over GF"2 of a completion of A. The conjecture is related to an old problem of proving super-linear lower bounds on the size of log-depth boolean circuits computing linear operators x@?Mx. The conjecture is also a generalization of a classical question about how much larger can non-linear codes be than linear ones. We prove some special cases of the conjecture and establish some structural properties of solution sets.