Swastik Kopparty

TL;DR: It is shown that degree-lifted Hermitian codes form a family of locally correctable codes over an alphabet that is significantly smaller than that obtained by Reed-Muller codes of similar constant rate, message length, and distance.

TL;DR: In this article, the authors extend the method of multiplicities to obtain tighter bounds on the size of the Kakeya set, which is tight to within a 2 + o(1) factor.

TL;DR: The list-decodability of multiplicity codes was studied in this paper, where it was shown that a polynomial over fields of prime order can be list decoded from a (1 R e ) fraction of errors in polynomially time.

Abstract: We consider the problem of testing if a given function f : F_2^n -> F_2 is close to any degree d polynomial in n variables, also known as the Reed-Muller testing problem. The Gowers norm is based on a natural 2^{d+1}-query test for this property. Alon et al. [AKKLR05] rediscovered this test and showed that it accepts every degree d polynomial with probability 1, while it rejects functions that are Omega(1)-far with probability Omega(1/(d 2^{d})). We give an asymptotically optimal analysis of this test, and show that it rejects functions that are (even only) Omega(2^{-d})-far with Omega(1)-probability (so the rejection probability is a universal constant independent of d and n). This implies a tight relationship between the (d+1)st Gowers norm of a function and its maximal correlation with degree d polynomials, when the correlation is close to 1. Our proof works by induction on n and yields a new analysis of even the classical Blum-Luby-Rubinfeld [BLR93] linearity test, for the setting of functions mapping F_2^n to F_2. The optimality follows from a tighter analysis of counterexamples to the "inverse conjecture for the Gowers norm" constructed by [GT09,LMS08]. Our result has several implications. First, it shows that the Gowers norm test is tolerant, in that it also accepts close codewords. Second, it improves the parameters of an XOR lemma for polynomials given by Viola and Wigderson [VW07]. Third, it implies a "query hierarchy" result for property testing of affine-invariant properties. That is, for every function q(n), it gives an affine-invariant property that is testable with O(q(n))-queries, but not with o(q(n))-queries, complementing an analogous result of [GKNR09] for graph properties.

TL;DR: This work shows that if the query complexity is relaxed to polynomial, then one can construct PCPs of linear length for circuit-SAT, and PCP of length O(tlog t) for any language in NTIME(t), and is the first constant-rate PCP construction that achieves constant soundness with nontrivial query complexity.