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On Rigid Matrices and U-Polynomials

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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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Authors and Affiliations

  1. Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, 69978, Tel Aviv, Israel

    Noga Alon

  2. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel

    Gil Cohen

Authors
  1. Noga Alon
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  2. Gil Cohen
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Correspondence to Gil Cohen.

Additional information

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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  • DOI: https://doi.org/10.1007/s00037-015-0112-9

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