Some structural properties of low-rank matrices related to computational complexity
Theoretical Computer Science - Selected papers in honor of Manuel Blum
A linear lower bound on the unbounded error probabilistic communication complexity
Journal of Computer and System Sciences - Complexity 2001
Extracting Randomness Using Few Independent Sources
SIAM Journal on Computing
On Locally Decodable Codes, Self-correctable Codes, and t-Private PIR
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Polarities, quasi-symmetric designs, and Hamada's conjecture
Designs, Codes and Cryptography
Perturbed identity matrices have high rank: Proof and applications
Combinatorics, Probability and Computing
Complexity Lower Bounds using Linear Algebra
Foundations and Trends® in Theoretical Computer Science
Blackbox Polynomial Identity Testing for Depth 3 Circuits
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
On Matrix Rigidity and Locally Self-Correctable Codes
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
On sums of locally testable affine invariant properties
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
SIAM Journal on Computing
The approximate rank of a matrix and its algorithmic applications: approximate rank
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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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 for m ≥ n, 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: Impossibility results for 2-query LCCs over large fields: 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 fields of characteristic zero or large characteristic regardless of the encoding length. Generalization of known results in combinatorial geometry: We prove a quantitative analog of the Sylvester-Gallai theorem: Let v1,...,vm be a set of points in Cd such that for every i ∈ [m] there exists at least δ m values of j ∈ [m] such that the line through vi,vj contains a third point in the set. We show that the dimension of v1,...,vm is at most O(1/δ2). Our results generalize to the high-dimensional case (replaceing lines with planes, etc.) and to the case where the points are colored (as in the Motzkin-Rabin Theorem).