On Rigid Matrices and Subspace Polynomials

We introduce a class of polynomials, which we call subspace 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 subspace 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. ∗Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. Email: [email protected]. Research supported in part by an ERC Advanced grant, by a USA-Israeli BSF grant and by the Israeli I-Core program. †Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. Email: [email protected]. Research supported by Israel Science Foundation (ISF) grant.