Improved Rank Bounds for Design Matrices and a New Proof of Kelly’s Theorem

Abstract

We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Barak et al. [Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes. Proceedings of the 43rd annual ACM symposium on Theory of computing, STOC 11, (ACM, NY 2011), 519-528] in which they were used to answer questions regarding point configurations. In this work, we derive near-optimal rank bounds for these matrices and use them to obtain asymptotically tight bounds in many of the geometric applications. As a consequence of our improved analysis, we also obtain a new, linear algebraic, proof of Kelly's theorem, which is the complex analog of the Sylvester-Gallai theorem.

Breaking the quadratic barrier for 3-LCC's over the reals

“…, S m and the fact that together they cover pairs from a quadratic number of triples. This should be contrasted with the results of [BDWY11,DSW12] which prove strong lower bounds for q-LCC's (for any constant q) in which every pair is in a bounded number of triples (these are called 'design' LCCs).…”

“…While this has not materialized yet, it motivated the invention of multiplicity codes by [KSY11] which are new LCCs of high rate, and turn out to yield optimal list-decodable codes as well [Kop12] . Finally, since the work of [DS06], LDCs and LCCs have played a role in understanding basic problems in Polynomial Identity Testing and established its connection to problems in Incidence Geometry, e.g [KS09, BDWY11,DSW12].…”

“…Strangely, while these vectors provide an LDC over every field, they fail to be an LCC except in F 2 . This gap was first explained in [BDWY11,DSW12] who showed that over the Real numbers (and indeed even finite fields of very large characteristic), LCCs simply do not exist! For every error-rate δ the dimension d for which such codes exist cannot exceed poly(1/δ).…”

“…For every error-rate δ the dimension d for which such codes exist cannot exceed poly(1/δ). The proofs in [BDWY11,DSW12] use a combination of geometric, analytic and linear-algebraic techniques, and give quantitative form to known qualitative point-line incidence theorems. Tighter bounds of n > p Ω(d) over finite fields of prime size p were proved in [BDSS11] using methods from arithmetic combinatorics, matching the trivial construction of taking all vectors in (F p ) d .…”

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“…Strangely, while these vectors provide an LDC over every field, they fail to be an LCC except over F 2 . This gap was first explained in [5,15] where the authors showed that over the real numbers (and indeed even over large enough finite fields), 2-LCCs simply do not exist! For every error-rate δ the dimension d for which such codes exist is finite, and cannot exceed poly(1/δ ).…”