Skip to main content

Tight lower bounds for linear 2-query LCCs over finite fields

Author(s): Bhattacharyya, Arnab; Dvir, Zeev; Saraf, Shubhangi; Shpilka, Amir

To refer to this page use:
Full metadata record
DC FieldValueLanguage
dc.contributor.authorBhattacharyya, Arnab-
dc.contributor.authorDvir, Zeev-
dc.contributor.authorSaraf, Shubhangi-
dc.contributor.authorShpilka, Amir-
dc.identifier.citationBhattacharyya, Arnab, Zeev Dvir, Shubhangi Saraf, and Amir Shpilka. "Tight lower bounds for linear 2-query LCCs over finite fields." Combinatorica 36, no. 1 (2016): pp. 1-36. doi:10.1007/s00493-015-3024-zen_US
dc.description.abstractA Locally Correctable Code (LCC) is an error correcting code that has a probabilistic self-correcting algorithm that, with high probability, can correct any coordinate of the codeword by looking at only a few other coordinates, even if a δ fraction of the coordinates is corrupted. LCCs are a stronger form of LDCs (Locally Decodable Codes) which have received a lot of attention recently due to their many applications and surprising constructions. In this work, we show a separation between linear 2-query LDCs and LCCs over finite fields of prime order. Specifically, we prove a lower bound of the form p Ω(δd) on the length of linear 2-query LCCs over F p , that encode messages of length d. Our bound improves over the known bound of 2Ω(δd) [8,10,6] which is tight for LDCs. Our proof makes use of tools from additive combinatorics which have played an important role in several recent results in theoretical computer science. We also obtain, as corollaries of our main theorem, new results in incidence geometry over finite fields. The first is an improvement to the Sylvester-Gallai theorem over finite fields [14] and the second is a new analog of Beck's theorem over finite fields. The paper also contains an appendix, written by Sergey Yekhanin, showing that there do exist nonlinear LCCs of size 2O(d) over F p , thus highlighting the importance of the linearity assumption for our result.en_US
dc.format.extent1 - 36en_US
dc.rightsAuthor's manuscripten_US
dc.titleTight lower bounds for linear 2-query LCCs over finite fieldsen_US
dc.typeJournal Articleen_US

Files in This Item:
File Description SizeFormat 
TightLowerBoundsLccsFiniteFieldsCombinatorica.pdf471.75 kBAdobe PDFView/Download

Items in OAR@Princeton are protected by copyright, with all rights reserved, unless otherwise indicated.