ECCC-Report TR19-056https://eccc.weizmann.ac.il/report/2019/056Comments and Revisions published for TR19-056en-usThu, 25 Apr 2019 12:02:06 +0300
Revision 1
| A Lower Bound for Relaxed Locally Decodable Codes |
Tom Gur,
Oded Lachish
https://eccc.weizmann.ac.il/report/2019/056#revision1A locally decodable code (LDC) C:{0,1}^k -> {0,1}^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(k^{1+ \gamma}) 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.Thu, 25 Apr 2019 12:02:06 +0300https://eccc.weizmann.ac.il/report/2019/056#revision1
Paper TR19-056
| A Lower Bound for Relaxed Locally Decodable Codes |
Tom Gur,
Oded Lachish
https://eccc.weizmann.ac.il/report/2019/056A locally decodable code (LDC) C:{0,1}^k -> {0,1}^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(k^{1+ \gamma}) 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 Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (STOC 2004).Sun, 14 Apr 2019 11:31:30 +0300https://eccc.weizmann.ac.il/report/2019/056