Min-Rank Conjecture for Log-Depth Circuits

TL;DR

This paper explores a conjecture linking matrix completion rank to the number of solutions in semi-linear systems over GF(2), with implications for circuit complexity and coding theory.

Contribution

The paper introduces a conjecture relating matrix rank to solution bounds in semi-linear systems and proves special cases, advancing understanding of circuit complexity and coding theory.

Findings

Proved special cases of the conjecture.

Established structural properties of solution sets.

Linked the conjecture to circuit lower bounds and coding theory.

Abstract

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.