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 or 1. A system of semi-linear equations over GF2 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 ·mr(A) solutions, where > 0 is an absolute constant and mr(A) is the smallest rank over GF2 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 y = 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.