Lower Bounds for 2-Query LCCs over Large Alphabet
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.
Locally correctable code
Private information retrieval
Szemerédi regularity lemma
30:1-30:20
Regular Paper
Arnab
Bhattacharyya
Arnab Bhattacharyya
Sivakanth
Gopi
Sivakanth Gopi
Avishay
Tal
Avishay Tal
10.4230/LIPIcs.APPROX-RANDOM.2017.30
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