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IMPROVED RANK BOUNDS FOR DESIGN MATRICES AND A NEW PROOF OF KELLY’S THEOREM

Part of: Theory of error-correcting codes and error-detecting codes Discrete geometry

Published online by Cambridge University Press:  28 March 2014

ZEEV DVIR ,
SHUBHANGI SARAF  and
AVI WIGDERSON
ZEEV DVIR
Affiliation:
Department of Computer Science and Department of Mathematics, Princeton University, USAzeev.dvir@gmail.com
SHUBHANGI SARAF
Affiliation:
Department of Computer Science and Department of Mathematics, Rutgers University, USA
AVI WIGDERSON
Affiliation:
School of Mathematics, Institute for Advanced Study, USA

Abstract

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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.

MSC classification

Secondary: 52C35: Arrangements of points, flats, hyperplanes 94B65: Bounds on codes
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e4
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2014

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