TY - GEN
T1 - Tight lower bounds for 2-query LCCs over finite fields
AU - Bhattacharyya, Arnab
AU - Dvir, Zeev
AU - Shpilka, Amir
AU - Saraf, Shubhangi
PY - 2011
Y1 - 2011
N2 - A 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 fraction δ of the coordinates are 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 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 double-struck F p, that encode messages of length d. Our bound improves over the known bound of 2 Ω(δd) [9], [12], [8] 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. Corollaries of our main theorem are new incidence geometry results 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.
AB - A 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 fraction δ of the coordinates are 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 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 double-struck F p, that encode messages of length d. Our bound improves over the known bound of 2 Ω(δd) [9], [12], [8] 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. Corollaries of our main theorem are new incidence geometry results 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.
KW - Sylvester-Gallai theorem
KW - additive combinatorics
KW - locally decodable codes
UR - http://www.scopus.com/inward/record.url?scp=84863301031&partnerID=8YFLogxK
U2 - https://doi.org/10.1109/FOCS.2011.28
DO - https://doi.org/10.1109/FOCS.2011.28
M3 - منشور من مؤتمر
SN - 9780769545714
T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
SP - 638
EP - 647
BT - Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011
T2 - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011
Y2 - 22 October 2011 through 25 October 2011
ER -