TY - GEN
T1 - Locally testable and locally correctable codes approaching the gilbert-varshamov bound
AU - Gopi, Sivakanth
AU - Kopparty, Swastik
AU - Oliveira, Rafael
AU - Ron-Zewi, Noga
AU - Saraf, Shubhangi
N1 - Publisher Copyright:
Copyright © by SIAM.
PY - 2017/1/1
Y1 - 2017/1/1
N2 - One of the most important open problems in the theory of error-correcting codes is to determine the tradeoff between the rate R and minimum distance δ of a binary code. The best known tradeoff is the Gilbert-Varshamov bound, and says that for every δϵ (0; 1=2), there are codes with minimum distance δ and rate R = RGV(δ) > 0 (for a certain simple function RGV(.)). In this paper we show that the Gilbert-Varshamov bound can be achieved by codes which support local error-detection and error- correction algorithms. Specifically, we show the following results. 1. Local Testing: For all δϵ (0; 1=2) and all R < RGV(δ), there exist codes with length n, rate R and minimum distance that are locally testable with quasipolylog(n) query complexity. 2. Local Correction: For all ϵ > 0, for all δ < 1=2 sufficiently large, and all R < (1 - ϵ)RGV(δ), there exist codes with length n, rate R and minimum distance that are locally correctable from δ 2 - o(1) fraction errors with O(nϵ) query complexity. Furthermore, these codes have an efficient randomized construction, and the local testing and local correction algorithms can be made to run in time polynomial in the query complexity. Our results on locally correctable codes also immediately give locally decodable codes with the same parameters. Our local testing result is obtained by combining Thommesen's random concatenation technique and the best known locally testable codes from [KMRS16]. Our local correction result, which is significantly more in- volved, also uses random concatenation, along with a number of further ideas: The Guruswami-Sudan-Indyk list decoding strategy for concatenated codes, Alon- Edmonds-Luby distance ampliffication, and the local list- decodability, local list-recoverability and local testability of Reed-Muller codes. Curiously, our final local correction algorithms go via local list-decoding and local testing al- gorithms; this seems to be the first time local testability is used in the construction of a locally correctable code.
AB - One of the most important open problems in the theory of error-correcting codes is to determine the tradeoff between the rate R and minimum distance δ of a binary code. The best known tradeoff is the Gilbert-Varshamov bound, and says that for every δϵ (0; 1=2), there are codes with minimum distance δ and rate R = RGV(δ) > 0 (for a certain simple function RGV(.)). In this paper we show that the Gilbert-Varshamov bound can be achieved by codes which support local error-detection and error- correction algorithms. Specifically, we show the following results. 1. Local Testing: For all δϵ (0; 1=2) and all R < RGV(δ), there exist codes with length n, rate R and minimum distance that are locally testable with quasipolylog(n) query complexity. 2. Local Correction: For all ϵ > 0, for all δ < 1=2 sufficiently large, and all R < (1 - ϵ)RGV(δ), there exist codes with length n, rate R and minimum distance that are locally correctable from δ 2 - o(1) fraction errors with O(nϵ) query complexity. Furthermore, these codes have an efficient randomized construction, and the local testing and local correction algorithms can be made to run in time polynomial in the query complexity. Our results on locally correctable codes also immediately give locally decodable codes with the same parameters. Our local testing result is obtained by combining Thommesen's random concatenation technique and the best known locally testable codes from [KMRS16]. Our local correction result, which is significantly more in- volved, also uses random concatenation, along with a number of further ideas: The Guruswami-Sudan-Indyk list decoding strategy for concatenated codes, Alon- Edmonds-Luby distance ampliffication, and the local list- decodability, local list-recoverability and local testability of Reed-Muller codes. Curiously, our final local correction algorithms go via local list-decoding and local testing al- gorithms; this seems to be the first time local testability is used in the construction of a locally correctable code.
UR - http://www.scopus.com/inward/record.url?scp=85016230516&partnerID=8YFLogxK
U2 - 10.1137/1.9781611974782.135
DO - 10.1137/1.9781611974782.135
M3 - Conference contribution
AN - SCOPUS:85016230516
T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
SP - 2073
EP - 2091
BT - 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
A2 - Klein, Philip N.
PB - Association for Computing Machinery
T2 - 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
Y2 - 16 January 2017 through 19 January 2017
ER -