Lachish, Oded and Gur, Tom (2019) A Lower Bound for Relaxed Locally Decodable Codes. In: ACM-SIAM Symposium on Discrete Algorithms (SODA20), 5-8 Jan 2020, Salt Lake City, U.S.. (In Press)
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Abstract
A locally decodable code (LDC) $C \colon \bitset^k \to \bitset^n$ is an error correcting code wherein individual bits of the message can be recovered by only querying a few bits of a noisy codeword. LDCs found a myriad of applications both in theory and in practice, ranging from probabilistically checkable proofs to distributed storage. However, despite nearly two decades of extensive study, the best known constructions of $O(1)$-query LDCs have super-polynomial blocklength. The notion of relaxed LDCs is a natural relaxation of LDCs, which aims to bypass the foregoing barrier by requiring local decoding of nearly all individual message bits, yet allowing decoding failure (but not error) on the rest. State of the art constructions of $O(1)$-query relaxed LDCs achieve blocklength $n = O\left(k^{1+ \gamma}\right)$ for an arbitrarily small constant $\gamma$. We prove a lower bound which shows that $O(1)$-query relaxed LDCs cannot achieve blocklength $n = k^{1+ o(1)}$. This resolves an open problem raised by Goldreich in 2004.
Metadata
| Item Type: | Conference or Workshop Item (Paper) |
|---|---|
| Keyword(s) / Subject(s): | Codes, local decoding, complexity, algorithms |
| School: | Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences |
| Depositing User: | Oded Lachish |
| Date Deposited: | 15 Jan 2020 12:45 |
| Last Modified: | 09 Aug 2023 12:47 |
| URI: | https://eprints.bbk.ac.uk/id/eprint/29537 |
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