Bridging Shannon and Hamming: List Error-Correction with Optimal Rate

Error-correcting codes tackle the fundamental problem of recovering from errors during data communication and storage. A basic issue in coding theory concerns the modeling of the channel noise. Shannon’s theory models the channel as a stochastic process with a known probability law. Hamming suggested a combinatorial approach where the channel causes worst-case errors subject only to a limit on the number of errors. These two approaches share a lot of common tools, however in terms of quantitative results, the classical results for worst-case errors were much weaker. We survey recent progress on list decoding, highlighting its power and generality as an avenue to construct codes resilient to worst-case errors with information rates similar to what is possible against probabilistic errors. In particular, we discuss recent explicit constructions of list-decodable codes with information-theoretically optimal redundancy that is arbitrarily close to the fraction of symbols that can be corrupted by worst-case errors.

[1]  Richard J. Lipton,et al.  A New Approach To Information Theory , 1994, STACS.

[2]  Leonid A. Levin,et al.  A hard-core predicate for all one-way functions , 1989, STOC '89.

[3]  Adam D. Smith Scrambling adversarial errors using few random bits, optimal information reconciliation, and better private codes , 2007, SODA '07.

[4]  Noga Alon,et al.  A linear time erasure-resilient code with nearly optimal recovery , 1996, IEEE Trans. Inf. Theory.

[5]  Enkatesan G Uruswami Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes , 2008 .

[6]  Ronitt Rubinfeld,et al.  Reconstructing algebraic functions from mixed data , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[7]  Venkatesan Guruswami,et al.  "Soft-decision" decoding of Chinese remainder codes , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[8]  Christopher Umans,et al.  Simple extractors for all min-entropies and a new pseudorandom generator , 2005, JACM.

[9]  W. W. Peterson,et al.  Encoding and error-correction procedures for the Bose-Chaudhuri codes , 1960, IRE Trans. Inf. Theory.

[10]  Silvio Micali,et al.  Optimal Error Correction Against Computationally Bounded Noise , 2005, TCC.

[11]  G. David Forney,et al.  Concatenated codes , 2009, Scholarpedia.

[12]  Venkatesan Guruswami,et al.  Improved decoding of Reed-Solomon and algebraic-geometric codes , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[13]  Imre Csiszár,et al.  The capacity of the arbitrarily varying channel revisited: Positivity, constraints , 1988, IEEE Trans. Inf. Theory.

[14]  M. Tsfasman,et al.  Modular curves, Shimura curves, and Goppa codes, better than Varshamov‐Gilbert bound , 1982 .

[15]  Venkatesan Guruswami,et al.  Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy , 2005, IEEE Transactions on Information Theory.

[16]  Vladimir M. Blinovsky,et al.  List decoding , 1992, Discrete Mathematics.

[17]  Venkatesan Guruswami,et al.  Limits to List Decoding Reed-Solomon Codes , 2006, IEEE Trans. Inf. Theory.

[18]  Ron M. Roth,et al.  Efficient decoding of Reed-Solomon codes beyond half the minimum distance , 2000, IEEE Trans. Inf. Theory.

[19]  Madhu Sudan,et al.  Highly Resilient Correctors for Polynomials , 1992, Inf. Process. Lett..

[20]  Venkatesan Guruswami,et al.  Explicit Capacity-achieving Codes for Worst-Case Additive Errors , 2009, ArXiv.

[21]  Venkatesan Guruswami,et al.  Combinatorial bounds for list decoding , 2002, IEEE Trans. Inf. Theory.

[22]  Madhu Sudan List decoding: algorithms and applications , 2000, SIGA.

[23]  Atri Rudra,et al.  Two Theorems in List Decoding , 2010, Electron. Colloquium Comput. Complex..

[24]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[25]  Venkatesan Guruswami,et al.  Linear-time encodable/decodable codes with near-optimal rate , 2005, IEEE Transactions on Information Theory.

[26]  Venkatesan Guruswami Cyclotomic function fields, Artin–Frobenius automorphisms, and list error correction with optimal rate , 2010 .

[27]  H. Stichtenoth,et al.  A low complexity algorithm for the construction of algebraic geometric codes better than the Gilbert-Varshamov bound , 2001, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252).

[28]  Alexander Vardy,et al.  Algebraic soft-decision decoding of Reed-Solomon codes , 2003, IEEE Trans. Inf. Theory.

[29]  Rüdiger L. Urbanke,et al.  Modern Coding Theory , 2008 .

[30]  Venkatesan Guruswami,et al.  On the List-Decodability of Random Linear Codes , 2010, IEEE Transactions on Information Theory.

[31]  Venkatesan Guruswami,et al.  Improved decoding of Reed-Solomon and algebraic-geometry codes , 1999, IEEE Trans. Inf. Theory.

[32]  Salil P. Vadhan,et al.  The unified theory of pseudorandomness , 2010 .

[33]  Richard W. Hamming,et al.  Error detecting and error correcting codes , 1950 .

[34]  Venkatesan Guruswami,et al.  List decoding algorithms for certain concatenated codes , 2000, STOC '00.

[35]  Salil P. Vadhan,et al.  The unified theory of pseudorandomness: guest column , 2007, SIGA.

[36]  Dana Ron,et al.  Chinese remaindering with errors , 1999, STOC '99.

[37]  Jaikumar Radhakrishnan,et al.  Subspace Polynomials and List Decoding of Reed-Solomon Codes , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[38]  Luca Trevisan,et al.  Some Applications of Coding Theory in Computational Complexity , 2004, Electron. Colloquium Comput. Complex..

[39]  Madhu Sudan,et al.  Extensions to the Method of Multiplicities, with Applications to Kakeya Sets and Mergers , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[40]  Erich Kaltofen,et al.  Polynomial Factorization 1987-1991 , 1992, LATIN.

[41]  Zeev Dvir,et al.  On the size of Kakeya sets in finite fields , 2008, 0803.2336.

[42]  G. David Forney,et al.  Generalized minimum distance decoding , 1966, IEEE Trans. Inf. Theory.

[43]  ChallengesPaul ZimmermannInria Lorrainezimmermann Polynomial Factorization , 1996 .

[44]  A. Rudra,et al.  Error correction up to the information-theoretic limit , 2009, CACM.

[45]  Henning Stichtenoth,et al.  Algebraic function fields and codes , 1993, Universitext.

[46]  Peter Elias,et al.  Error-correcting codes for list decoding , 1991, IEEE Trans. Inf. Theory.

[47]  Venkatesan Guruswami,et al.  Algorithmic Results in List Decoding , 2006, Found. Trends Theor. Comput. Sci..

[48]  Robert J. McEliece,et al.  On the decoder error probability for Reed-Solomon codes , 1986, IEEE Trans. Inf. Theory.

[49]  H. Stichtenoth,et al.  A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound , 1995 .

[50]  Madhu Sudan,et al.  Decoding of Reed Solomon Codes beyond the Error-Correction Bound , 1997, J. Complex..

[51]  Venkatesan Guruswami,et al.  List decoding of error correcting codes , 2001 .

[52]  Peter Elias,et al.  List decoding for noisy channels , 1957 .

[53]  V. D. Goppa Codes on Algebraic Curves , 1981 .

[54]  Venkatesan Guruswami Iterative Decoding of Low-Density Parity Check Codes , 2006, Bull. EATCS.

[55]  Alexander Vardy,et al.  Correcting errors beyond the Guruswami-Sudan radius in polynomial time , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).