eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-08-11
30:1
30:20
10.4230/LIPIcs.APPROX-RANDOM.2017.30
article
Lower Bounds for 2-Query LCCs over Large Alphabet
Bhattacharyya, Arnab
Gopi, Sivakanth
Tal, Avishay
A locally correctable code (LCC) is an error correcting code that allows correction of any arbitrary coordinate of a corrupted codeword by querying only a few coordinates. We show that any 2-query locally correctable code C:{0,1}^k -> Sigma^n that can correct a constant fraction of corrupted symbols must have n >= exp(k/\log|Sigma|) under the assumption that the LCC is zero-error. We say that an LCC is zero-error if there exists a non-adaptive corrector algorithm that succeeds with probability 1 when the input is an uncorrupted codeword. All known constructions of LCCs are zero-error.
Our result is tight upto constant factors in the exponent. The only previous lower bound on the length of 2-query LCCs over large alphabet was Omega((k/log|\Sigma|)^2) due to Katz and Trevisan (STOC 2000). Our bound implies that zero-error LCCs cannot yield 2-server private information retrieval (PIR) schemes with sub-polynomial communication. Since there exists a 2-server PIR scheme with sub-polynomial communication (STOC 2015) based on a zero-error 2-query locally decodable code (LDC), we also obtain a separation between LDCs and LCCs over large alphabet.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol081-approx-random2017/LIPIcs.APPROX-RANDOM.2017.30/LIPIcs.APPROX-RANDOM.2017.30.pdf
Locally correctable code
Private information retrieval
Szemerédi regularity lemma