ECCC-Report TR16-189https://eccc.weizmann.ac.il/report/2016/189Comments and Revisions published for TR16-189en-usTue, 22 Nov 2016 04:06:17 +0200
Paper TR16-189
| Lower bounds for 2-query LCCs over large alphabet |
Sivakanth Gopi,
Arnab Bhattacharyya
https://eccc.weizmann.ac.il/report/2016/189A 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 zero-error $2$-query locally correctable code $\mathcal{C}: \{0,1\}^k \to \Sigma^n$ that can correct a constant fraction of corrupted symbols must have $n \geq \exp(k/\log|\Sigma|)$. 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 $\Omega\left((k/\log|\Sigma|)^2\right)$ 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.
For our proof of the result, we need a new decomposition lemma for directed graphs that may be of independent interest. Given a dense directed graph $G$, our decomposition uses the directed version of Szemer\'edi regularity lemma due to Alon and Shapira (STOC 2003) to partition almost all of $G$ into a constant number of subgraphs which are either edge-expanding or empty.Tue, 22 Nov 2016 04:06:17 +0200https://eccc.weizmann.ac.il/report/2016/189