Soft Decoding, Dual BCH Codes, and Better List-Decodable e-Biased Codes

We construct binary linear codes that are efficiently list- decodable up to a fraction (1/2 - epsiv) of errors. The codes encode k bits into n = poly(k/epsiv) bits and are constructible and list-decodable in time polynomial in k and 1/epsiv (in particular, in our results epsiv need not be constant and can even be polynomially small in n). Our results give the best known polynomial dependence of n on k and 1/epsiv for such codes. Specifically, we are able to achieve n les O(k3/epsiv3+gamma) or, if a linear dependence on k is required, n les O (k/epsiv5+gamma), where gamma > 0 is an arbitrary constant. The best previously known constructive bounds in this setting were n les O(k2/epsiv4) and n les O(k/ epsiv6). Non-constructively, a random linear encoding of length n = O(k/epsiv2) suffices, but no sub-exponential algorithm is known for list decoding random codes. Our construction with a cubic dependence on epsiv is obtained by concatenating the recent Parvaresh-Vardy (PV) codes with dual BCH codes, and crucially exploits the soft decoding algorithm for PV codes. This result yields better hardness results for the problem of approximating NP witnesses in the model of Kumar and Sivakumar. Our result with the linear dependence on k is based on concatenation of the PV code with an arbitrary inner code of good minimum distance. In addition to being a basic question in coding theory, codes that are list-decodable from a fraction (1/2 - epsiv) of errors for epsiv rarr 0 have found many uses in complexity theory. In addition, our codes have the property that all nonzero codewords have relative Hamming weights in the range (1/2 - epsiv, 1/2 + epsiv); this epsiv-biased property is a fundamental notion in pseudorandomness.

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

[2]  Richard A. Brualdi,et al.  Greedy Codes , 1993, J. Comb. Theory, Ser. A.

[3]  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).

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

[5]  Christopher Umans,et al.  Simple extractors for all min-entropies and a new pseudo-random generator , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[6]  Noga Alon,et al.  Construction Of Asymptotically Good Low-rate Error-correcting Codes Through Pseudo-random Graphs , 1991, Proceedings. 1991 IEEE International Symposium on Information Theory.

[7]  Venkatesan Guruswami,et al.  Explicit capacity-achieving list-decodable codes , 2005, STOC.

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

[9]  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).

[10]  Elchanan Mossel,et al.  On the complexity of approximating the VC dimension , 2001, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[11]  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).

[12]  Luca Trevisan,et al.  Pseudorandom generators without the XOR Lemma , 1999, Electron. Colloquium Comput. Complex..

[13]  Moni Naor,et al.  Small-bias probability spaces: efficient constructions and applications , 1990, STOC '90.

[14]  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).

[15]  Noga Alon,et al.  Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs , 1992, IEEE Trans. Inf. Theory.

[16]  A. Rudra,et al.  List decoding and property testing of error-correcting codes , 2007 .

[17]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[18]  Venkatesan Guruswami,et al.  Decoding concatenated codes using soft information , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[19]  Venkatesan Guruswami.  8 List Decoding of Concatenated Codes , 2004 .

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

[21]  Amnon Ta-Shma,et al.  Constructing Small-Bias Sets from Algebraic-Geometric Codes , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[22]  Simon Litsyn,et al.  Almost orthogonal linear codes are locally testable , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[23]  Marcos A. Kiwi.  Testing and Weight Distributions of Dual Codes , 1997, Electron. Colloquium Comput. Complex..

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

[25]  Ralf Koetter.  On optimal weight assignmnents for multivariate interpolation list-decoding , 2006, 2006 IEEE Information Theory Workshop - ITW '06 Punta del Este.

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

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

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

[29]  J. Kuelbs.  Probability on Banach spaces , 1978 .

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

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

[32]  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).

[33]  E. Berlekamp.  Factoring polynomials over large finite fields* , 1971, SYMSAC '71.

[34]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.