Superquadratic Lower Bound for 3-Query Locally Correctable Codes over the Reals

We prove that 3-query linear locally correctable codes of dimension d over the reals require block length n > d2+α for some fixed, positive α > 0. Geometrically, this means that if n vectors in Rd are such that each vector is spanned by a linear number of disjoint triples of others, then it must be that n > d2+α . This improves the known quadratic lower bounds (e. g., Kerenidis–de Wolf (2004), Woodruff (2007)). While the improvement is modest, we expect that the new techniques introduced in this article will be useful for further progress on lower bounds of locally correctable and decodable codes with more than 2 queries, possibly over other fields as well. Several of the new ideas in the proof work over every field. At a high level, our proof has two parts, clustering and random restriction. The clustering step uses a powerful theorem of Barthe from convex geometry. It can be used (after preprocessing our LCC to be balanced), to apply a basis change (and rescaling) of the vectors, so that the resulting unit vectors become nearly isotropic. This together with the fact that any LCC must have many “correlated” pairs of points, lets us deduce that the An extended abstract of this paper appeared in the Proceedings of the Forty-sixth Annual ACM Symposium on Theory of Computing 2014 [14]. ∗Supported by NSF CAREER award DMS-1451191 and NSF grant CCF-1523816. †Supported by NSF grant CCF-1350572. ‡Supported by NSF grant CCF-1412958. ACM Classification: E.4 AMS Classification: 94B65, 52C35