Venkatesan Guruswami
Piotr Indyk
ABSTRACT
We present the first construction of error-correcting codes which can be (list) decoded from a noise fraction arbitrarily close to 1 in linear time. Specifically, we present an explicit construction of codes which can be encoded in linear time as well as list decoded in linear time from a fraction (1-ε) of errors for arbitrary ε > 0. The rate and alphabet size of the construction are constants that depend only on ε. Our construction involves devising a new combinatorial approach to list decoding, in contrast to all previous approaches which relied on the power of decoding algorithms for algebraic codes like Reed-Solomon codes.Our result implies that it is possible to have, and in fact explicitly specifies, a coding scheme for arbitrarily large noise thresholds with only constant redundancy in the encoding and constant amount of work (at both the sending and receiving ends) for each bit of information to be communicated. Such a result was known for certain probabilistic error models, and here we show that this is possible under the stronger adversarial noise model as well.
- Michael Alekhnovich. Linear Diophantine equations over polynomials and soft decoding of Reed-Solomon codes. Proceedings of FOCS 2002, pages 439--448. Google Scholar
Digital Library
- Noga Alon, Jehoshua Bruck, Joseph Naor, Moni Naor, and Ronny Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Transactions on Information Theory, 38:509--516, 1992.Google ScholarDigital Library
- Noga Alon and Fan R. K. Chung. Explicit construction of linear sized tolerant networks. Discrete Math. 72 (1988), pp. 15--19. Google ScholarDigital Library
- Noga Alon, Jeff Edmonds and Michael Luby. Linear time erasure codes with nearly optimal recovery. Proceedings of FOCS'95. Google ScholarDigital Library
- Sigal Ar, Richard J. Lipton, Ronitt Rubinfeld and Madhu Sudan. Reconstructing Algebraic Functions from Mixed Data. SIAM Journal on Computing, 28(2): 487--510, April 1999. Google ScholarDigital Library
- Daniel Bleichenbacher and P. Nguyen, Noisy Polynomial Interpolation and Noisy Chinese Remaindering. Proceedings of EUROCRYPT '2000, LNCS vol. 1807, Springer-Verlag, pp. 53--69, 2000. Google ScholarDigital Library
- Peter Elias. List decoding for noisy channels. Wescon Convention Record, Part 2, Institute of Radio Engineers (now IEEE), pp. 94--104, 1957.Google Scholar
- G. L. Feng. Two Fast Algorithms in the Sudan Decoding Procedure. Proceedings of the 37th Annual Allerton Conference on Communication, Control and Computing, pp. 545--554, 1999.Google Scholar
- Venkatesan Guruswami. List Decoding of Error-Correcting Codes. Ph.D thesis, Massachusetts Institute of Technology, August 2001. Google ScholarDigital Library
- Venkatesan Guruswami and Piotr Indyk. Expander-based constructions of efficiently decodable codes. Proceedings of the 42nd Annual Symposium on Foundations of Computer Science, Las Vegas, NV, October 2001, pages 658--667. Google ScholarDigital Library
- Venkatesan Guruswami and Piotr Indyk. Near-optimal Linear-Time Codes for Unique Decoding and New List-Decodable Codes Over Smaller Alphabets. Proceedings of STOC'02, pages 812-821. Google ScholarDigital Library
- Venkatesan Guruswami and Madhu Sudan. Improved decoding of Reed-Solomon and algebraic geometric codes. IEEE Transactions on Information Theory, 45:1757--1767, 1999. Google ScholarDigital Library
- Venkatesan Guruswami and Madhu Sudan. List decoding algorithms for certain concatenated codes. Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pages 181--190, 2000. Google ScholarDigital Library
- J. Kuczynski and H. Wozniakowski. Estimating the largest eigenvalue by the power and Lanczos algorithms with a random start. SIAM J. Matrix Anal. Appl., 15 (1994), pp. 672--691. Google ScholarDigital Library
- Alex Lubotzky, R. Phillips, and Peter Sarnak. Ramanujan graphs. Combinatorica, 8(3):261--277, 1988.Google ScholarCross Ref
- Ronny Roth and Gitit Ruckenstein. Efficient decoding of Reed-Solomon codes beyond half the minimum distance. IEEE Transactions on Information Theory, 46(1):246--257, January 2000. Google ScholarDigital Library
- Daniel Spielman. Computationally Efficient Error-Correcting Codes and Holographic Proofs. Ph.D thesis, Massachusetts Institute of Technology, June 1995. Google ScholarDigital Library
- Daniel Spielman. Linear-time encodable and decodable error-correcting codes. IEEE Transactions on Information Theory, 42(6):1723--1732, 1996. Google ScholarCross Ref
- Daniel Spielman. The Complexity of Error-Correcting Codes. 11th International Symposium on Fundamentals of Computation Theory, Lecture Notes in Computer Science # 1279, pp. 67--84; Krakow, Poland, September 1997. Google ScholarDigital Library
- Madhu Sudan. Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180--193, 1997. Google ScholarDigital Library
- Amnon Ta-Shma and David Zuckerman. Extractor Codes. Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pages 193--199, 2001. Google ScholarDigital Library
- J. M. Wozencraft. List Decoding. Quarterly Progress Report, Research Laboratory of Electronics, MIT, Vol. 48 (1958), pp. 90--95.Google Scholar
Index Terms
- Linear time encodable and list decodable codes
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