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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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
- Matrix rigidity
- small-bias sets
- unbalanced expanders