List decoding: algorithms and applications

Over the years coding theory and complexity theory have benefited from a number of mutually enriching connections. This article focuses on a new connection that has emerged between the two topics in the recent years. This connection is centered around the notion of “list-decoding” for error-correcting codes. In this survey we describe the list-decoding problem, the algorithms that have been developed, and a diverse collection of applications within complexity theory.

[1]  Madhu Sudan,et al.  Improved Low-Degree Testing and its Applications , 1997, STOC '97.

[2]  Richard J. Lipton,et al.  Computing from partial solutions , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[3]  Dana Ron,et al.  Chinese remaindering with errors , 2000, IEEE Trans. Inf. Theory.

[4]  Luca Trevisan,et al.  Pseudorandom generators without the XOR Lemma (extended abstract) , 1999, STOC '99.

[5]  D. Sivakumar,et al.  On membership comparable sets , 1998, Proceedings. Thirteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat. No.98CB36247).

[6]  Luca Trevisan,et al.  Construction of extractors using pseudo-random generators (extended abstract) , 1999, STOC '99.

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

[8]  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).

[9]  Noam Nisan,et al.  BPP has subexponential time simulations unlessEXPTIME has publishable proofs , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

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

[11]  Amin Shokrollahi,et al.  List Decoding of Algebraic-Geometric Codes , 1999, IEEE Trans. Inf. Theory.

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

[13]  Manuel Blum,et al.  How to generate cryptographically strong sequences of pseudo random bits , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

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

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

[16]  Avi Wigderson,et al.  P = BPP if E requires exponential circuits: derandomizing the XOR lemma , 1997, STOC '97.

[17]  Dan Boneh,et al.  Finding smooth integers in short intervals using CRT decoding , 2000, STOC '00.

[18]  Oded Goldreich,et al.  RSA and Rabin Functions: Certain Parts are as Hard as the Whole , 1988, SIAM J. Comput..

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

[20]  Paul S. Wang,et al.  Polynomial Factorization Sharp Bounds, Efficient Algorithms , 1993, J. Symb. Comput..

[21]  Russell Impagliazzo,et al.  Hard-core distributions for somewhat hard problems , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[22]  Ronitt Rubinfeld,et al.  Reconstructing Algebraic Functions from Mixed Data , 1998, SIAM J. Comput..

[23]  Ronitt Rubinfeld,et al.  Learning polynomials with queries: The highly noisy case , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[24]  Joan Feigenbaum,et al.  Hiding Instances in Multioracle Queries , 1990, STACS.

[25]  Jin-Yi Cai,et al.  On the Hardness of Permanent , 1999, STACS.

[26]  Ravi Kumar,et al.  Proofs, codes, and polynomial-time reducibilities , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[27]  Ran Raz,et al.  Extracting all the randomness and reducing the error in Trevisan's extractors , 1999, STOC '99.

[28]  Avi Wigderson,et al.  Near-optimal conversion of hardness into pseudo-randomness , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[29]  Noam Nisan,et al.  Hardness vs Randomness , 1994, J. Comput. Syst. Sci..

[30]  Richard J. Lipton,et al.  New Directions In Testing , 1989, Distributed Computing And Cryptography.