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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 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 i-th coordinate of which can only depend on
variables corresponding to *-entries in the i-th row of A. We conjecture that
no such system can have more than 2^{n-c\cdot mr(A)} solutions, where c>0 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.Comment: 22 pages, to appear in: J. Comput.Syst.Sci
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