A (q,k,t)-design matrix is an m x n matrix whose pattern of zeros non-zeros satisfies the following design-like condition: each row has at most q non-zeros, each column has at least k non-zeros and the supports of every two columns intersect in at most t rows. We prove that the rank of any (q,k,t)-design matrix over a field of characteristic zero (or sufficiently large finite characteristic) is at least n - (qtn 2k)^2 . Using this result we derive the following applications: (1) Impossibility results for 2-query LCCs over the complex numbers: A 2-query locally correctable code (LCC) is an error correcting code in which every codeword coordinate can be recovered, probabilistically, by reading at most two other code positions. Such codes have numerous applications and constructions (with exponential encoding length) are known over finite fields of small characteristic. We show that infinite families of such linear 2-query LCCs do not exist over the complex numbers. (2) Generalization of results in combinatorial geometry: We prove a quantitative analog of the Sylvester-Gallai theorem: Let @math be a set of points in @math such that for every @math there exists at least @math values of @math such that the line through @math contains a third point in the set. We show that the dimension of @math is at most @math . Our results generalize to the high dimensional case (replacing lines with planes, etc.) and to the case where the points are colored (as in the Motzkin-Rabin Theorem).
The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i≠j) is nonzero whenever i,j is an edge in G and is zero otherwise. This paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues.