On Rigid Matrices and U-Polynomials

Abstract

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.

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Correspondence to Gil Cohen.

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Research supported in part by an ERC Advanced grant, by a USA-Israeli BSF grant and by the Israeli I-Core program.

Research supported by Israel Science Foundation (ISF) grant.

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Alon, N., Cohen, G. On Rigid Matrices and U-Polynomials. comput. complex. 24, 851–879 (2015). https://doi.org/10.1007/s00037-015-0112-9

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Keywords

  • Matrix rigidity
  • small-bias sets
  • unbalanced expanders

Subject classification

  • 68Q17
  • 68R05