On Rigid Matrices and U-Polynomials

We introduce a class of polynomials, which we call U - polynomials , and show that the problem of explicitly constructing a rigid matrix can be reduced to the problem of explicitly constructing a small hitting set for this class. We prove that small-bias sets are hitting sets for the class of U -pol...

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Bibliographic Details
Published in:	Computational complexity Vol. 24; no. 4; pp. 851 - 879
Main Authors:	Alon, Noga, Cohen, Gil
Format:	Journal Article
Language:	English
Published:	Cham Springer International Publishing 01.12.2015
Subjects:	Algorithm Analysis and Problem Complexity Computational Mathematics and Numerical Analysis Computer Science small-bias sets Matrix rigidity unbalanced expanders 68Q17 68R05
ISSN:	1016-3328, 1420-8954
Online Access:	Get full text
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Summary:	We introduce a class of polynomials, which we call U - polynomials , and show that the problem of explicitly constructing a rigid matrix can be reduced to the problem of explicitly constructing a small hitting set for this class. We prove that small-bias sets are hitting sets for the class of U -polynomials, though their size is larger than desired. Furthermore, we give two alternative proofs for the fact that small-bias sets induce rigid matrices. Finally, we construct rigid matrices from unbalanced expanders, with essentially the same size as the construction via small-bias sets.
ISSN:	1016-3328 1420-8954
DOI:	10.1007/s00037-015-0112-9