Min-rank conjecture for log-depth circuits

Abstract

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 G F 2 has the form M x = 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 − ϵ ⋅ mr ( A ) solutions, where ϵ > 0 is an absolute constant and mr ( A ) is the smallest rank over G F 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 ↦ M x . 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.