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

On an infinite sequence of improving Boolean bases

AbstractWe consider complexity of formulas for Boolean functions in finite complete bases. It is shown that, as regards complexity, the basis of all (k+1)-ary functions is essentially better than the basis of all k-ary functions for all k⩾2

FLEXIBLE POLYMER-BASED MICROWAVE PASSIVE DEVICES AND ANTENNAS

Ph.DDOCTOR OF PHILOSOPHY (FOE