Breaking the quadratic barrier for 3-LCC's over the Reals

Zeev Dvir, Shubhangi Saraf, Avi Wigderson

Research output: Chapter in Book/Report/Conference proceedingConference contribution

15 Scopus citations

Abstract

We prove that 3-query linear locally correctable codes over the Reals of dimension d require block length n > d2+λ for some fixed, positive λ > 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+λ. This improves the known quadratic lower bounds (e.g. [20, 28]). While a modest improvement, we expect that the new techniques introduced in this work will be useful for further progress on lower bounds of locally correctable and decodable codes with more than 2 queries. At a high level, our proof has two parts, clustering and random restriction. The clustering step uses a powerful geometric theorem of Barthe which finds a basis change (and rescaling) putting the code in nearly isotropic position. This together with the fact that any LCC must have many 'correlated' pairs of points, lets us deduce that the vectors must have a surprisingly strong geometric clustering, and hence also combinatorial clustering with respect to the spanning triples. In the restriction step, we devise a new variant of the dimension reduction technique used in previous lower bounds, which is able to take advantage of the combinatorial clustering structure above. The analysis of our random projection method reduces to a simple (weakly) random graph process, and works over any field.

Original languageEnglish (US)
Title of host publicationSTOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery
Pages784-793
Number of pages10
ISBN (Print)9781450327107
DOIs
StatePublished - Jan 1 2014
Event4th Annual ACM Symposium on Theory of Computing, STOC 2014 - New York, NY, United States
Duration: May 31 2014Jun 3 2014

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Other

Other4th Annual ACM Symposium on Theory of Computing, STOC 2014
CountryUnited States
CityNew York, NY
Period5/31/146/3/14

All Science Journal Classification (ASJC) codes

  • Software

Cite this

Dvir, Z., Saraf, S., & Wigderson, A. (2014). Breaking the quadratic barrier for 3-LCC's over the Reals. In STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing (pp. 784-793). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/2591796.2591818