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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

References

Alon, N., ‘Perturbed identity matrices have high rank: proof and applications’, Combin. Probab. Comput. 18 (2009), 3–15.Google Scholar
Barak, B., Dvir, Z., Wigderson, A. and Yehudayoff, A., ‘Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes’, inProceedings of the 43rd Annual ACM Symposium on Theory of Computing, STOC ’11 (ACM, New York, NY, USA, 2011), 519–528.Google Scholar
Barkol, O., Ishai, Y. and Weinreb, E., ‘On locally decodable codes, self-correctable codes, and t-private PIR’, inAPPROX’07/RANDOM’07: Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques (Springer-Verlag, Berlin, Heidelberg, 2007), 311–325.Google Scholar
Bhattacharyya, A., Dvir, Z., Shpilka, A. and Saraf, S., Tight lower bounds for 2-query LCCs over finite fields, Proc. of FOCS 2011, 2011, 638–647.Google Scholar
Bonnice, W. and Edelstein, M., ‘Flats associated with finite sets in $\mathbb{P}^d$ ’, Niew. Arch. Wisk. 15 (1967), 11–14.Google Scholar
Borwein, P. and Moser, W. O. J., ‘A survey of Sylvester’s problem and its generalizations’, Aequationes Math. 40 (1990), 111–135.Google Scholar
Dvir, Z., Incidence theorems and their applications, 2012. Preprint available at http://arxiv.org/abs/1208.5073.Google Scholar
Dvir, Z. and Shpilka, A., ‘Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits’, SIAM J. Comput. 36 (2006), 1404–1434.Google Scholar
Elkies, N. D., Pretorius, L. M. and Swanepoel, K. J., ‘Sylvester–Gallai theorems for complex numbers and quaternions’, Discrete Comput. Geom. 35 (2006), 361–373.Google Scholar
Erdos, P., Problems for solution: 4065–4069, 1943.Google Scholar
Fallat, S. M. and Hogben, L., ‘The minimum rank of symmetric matrices described by a graph: a survey’, Linear Algebra Appl. 426 (2007), 558–582.Google Scholar
Forster, J., ‘A linear lower bound on the unbounded error probabilistic communication complexity’, J. Comput. Syst. Sci. 65 (2002), 612–625.Google Scholar
Hamada, N., ‘On the $p$ -rank of the incidence matrix of a balanced or partially balanced incomplete block design and its application to error correcting codes’, Hiroshima Math. J. 3 (1973), 154–226.Google Scholar
Hansen, S., ‘A generalization of a theorem of Sylvester on the lines determined by a finite point set’, Math. Scand. 16 (1965), 175–180.Google Scholar
Hilton, A. J. W., ‘On double diagonal and cross latin squares’, J. Lond. Math. Soc. s2–s6 (1973), 679–689.CrossRefGoogle Scholar
Jungnickel, D. and Tonchev, V. D., ‘Polarities, quasi-symmetric designs, and Hamada’s conjecture’, Des. Codes Cryptogr. 51 (2009), 131–140.Google Scholar
Kayal, N. and Saraf, S., ‘Blackbox polynomial identity testing for depth 3 circuits’, inFOCS’09: Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science (IEEE Computer Society, Washington, DC, USA, 2009), 198–207.CrossRefGoogle Scholar
Kelly, L. M., ‘A resolution of the Sylvester–Gallai problem of J. P. Serre’, Discrete Comput. Geom. 1 (1986), 101–104.Google Scholar
Linial, N., Samorodnitsky, A. and Wigderson, A., ‘A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents’, Combinatorica 20 (2000), 545–568.Google Scholar
Lokam, S. V., ‘Complexity lower bounds using linear algebra’, Found. Trends Theor. Comput. Sci. 4 (2009), 1–155.Google Scholar
Melchior, E., ‘Uber vielseite der projektive ebene’, Deutsche Math. 5 (1940), 461–475.Google Scholar
Razborov, A. A. and Sherstov, A. A., ‘The sign-rank of $AC^0$ ’, inFOCS ’08: Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science (IEEE Computer Society, Washington, DC, USA, 2008), 57–66.Google Scholar
Rothblum, U. and Schneider, H., ‘Scaling of matrices which have pre-specified row sums and column sums via optimization’, Linear Algebra Appl. 114/115 (1989), 737–764.Google Scholar
Saxena, N. and Seshadhri, C., From Sylvester-gallai configurations to rank bounds: Improved black-box identity test for depth-3 circuits, Foundations of Computer Science, Annual IEEE Symposium on 0:21–29.Google Scholar
Sinkhorn, R., ‘A relationship between arbitrary positive matrices and doubly stochastic matrices’, Ann. Math. Stat. 35 (1964), 876–879.Google Scholar
Sylvester, J. J., ‘Mathematical question 11851’, Educ. Times 59 (1893), 98.Google Scholar
Tao, T. and Vu, V. H., Additive Combinatorics, Cambridge Studies in Advanced Mathematics, Number v. 13 (Cambridge University Press, 2006).Google Scholar
Valiant, L. G., Graph-theoretic arguments in low-level complexity, MFCS, 1977, 162–176.Google Scholar