Improved Decoding of Folded Reed-Solomon and Multiplicity Codes

In this work, we show new and improved error-correcting properties of folded Reed-Solomon codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the sources of recent advances in coding theory. Folded Reed-Solomon codes were the first explicit constructions of codes known to achieve list-decoding capacity; multivariate multiplicity codes were the first constructions of high-rate locally correctable codes; and univariate multiplicity codes are also known to achieve list-decoding capacity. However, previous analyses of the error-correction properties of these codes did not yield optimal results. In particular, in the list-decoding setting, the guarantees on the list-sizes were polynomial in the block length, rather than constant; and for multivariate multiplicity codes, local list-decoding algorithms could not go beyond the Johnson bound. In this paper, we show that Folded Reed-Solomon codes and multiplicity codes are in fact better than previously known in the context of list decoding and local list-decoding. More precisely, we first show that Folded RS codes achieve list-decoding capacity with constant list sizes, independent of the block length; and that high-rate univariate multiplicity codes can also be list-recovered with constant list sizes. Using our result on univariate multiplicity codes, we show that multivariate multiplicity codes are high-rate, locally list-recoverable codes. Finally, we show how to combine the above results with standard tools to obtain capacity achieving locally list decodable codes with query complexity significantly lower than was known before.

[1]  Brett Hemenway,et al.  Local List Recovery of High-Rate Tensor Codes and Applications , 2020, SIAM J. Comput..

[2]  Atri Rudra,et al.  Efficiently decodable non-adaptive group testing , 2010, SODA '10.

[3]  Richard J. Lipton Efficient Checking of Computations , 1990, STACS.

[4]  Madhu Sudan,et al.  Improved low-degree testing and its applications , 1997, STOC '97.

[5]  Venkatesan Guruswami,et al.  Linear-Time List Decoding in Error-Free Settings: (Extended Abstract) , 2004, ICALP.

[6]  Venkatesan Guruswami,et al.  Linear-Algebraic List Decoding for Variants of Reed–Solomon Codes , 2013, IEEE Transactions on Information Theory.

[7]  Or Meir,et al.  High-rate locally-correctable and locally-testable codes with sub-polynomial query complexity , 2016, STOC.

[8]  Brett Hemenway,et al.  Local List Recovery of High-Rate Tensor Codes & Applications , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

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

[10]  Venkatesan Guruswami,et al.  Explicit subspace designs , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[11]  Venkatesan Guruswami,et al.  List decoding reed-solomon, algebraic-geometric, and gabidulin subcodes up to the singleton bound , 2013, STOC '13.

[12]  Shachar Lovett,et al.  Subspace evasive sets , 2012, STOC '12.

[13]  Alan Guo,et al.  List-Decoding Algorithms for Lifted Codes , 2016, IEEE Transactions on Information Theory.

[14]  Venkatesan Guruswami,et al.  Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy , 2008, IEEE Transactions on Information Theory.

[15]  Venkatesan Guruswami,et al.  Near-optimal linear-time codes for unique decoding and new list-decodable codes over smaller alphabets , 2002, STOC '02.

[16]  Rafail Ostrovsky,et al.  Local correctability of expander codes , 2015, Inf. Comput..

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

[18]  Brett Hemenway,et al.  Linear-time list recovery of high-rate expander codes , 2018, Inf. Comput..

[19]  Eyal Kushilevitz,et al.  Learning decision trees using the Fourier spectrum , 1991, STOC '91.

[20]  Venkatesan Guruswami List Decoding of Error-Correcting Codes (Winning Thesis of the 2002 ACM Doctoral Dissertation Competition) , 2004, Lecture Notes in Computer Science.

[21]  Swastik Kopparty,et al.  Decoding Reed-Muller codes over product sets , 2016, Computational Complexity Conference.

[22]  Noga Alon,et al.  Linear time erasure codes with nearly optimal recovery , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[23]  Luca Trevisan,et al.  Pseudorandom generators without the XOR Lemma , 1998, Electron. Colloquium Comput. Complex..

[24]  Venkatesan Guruswami,et al.  Linear time encodable and list decodable codes , 2003, STOC '03.

[25]  Jonathan Katz,et al.  On the efficiency of local decoding procedures for error-correcting codes , 2000, STOC '00.

[26]  Shubhangi Saraf,et al.  Locally testable and Locally correctable Codes Approaching the Gilbert-Varshamov Bound , 2016, Electron. Colloquium Comput. Complex..

[27]  S. Vadhan Pseudorandomness II , 2009 .

[28]  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).

[29]  Leonid A. Levin,et al.  Checking computations in polylogarithmic time , 1991, STOC '91.

[30]  Shubhangi Saraf,et al.  Noisy Interpolation of Sparse Polynomials, and Applications , 2011, 2011 IEEE 26th Annual Conference on Computational Complexity.

[31]  Alan Guo,et al.  New affine-invariant codes from lifting , 2013, ITCS '13.

[32]  Or Meir,et al.  High-Rate Locally Correctable and Locally Testable Codes with Sub-Polynomial Query Complexity , 2017, J. ACM.

[33]  Atri Rudra,et al.  Efficiently Decodable Compressed Sensing by List-Recoverable Codes and Recursion , 2012, STACS.

[34]  Venkatesan Guruswami,et al.  Folded codes from function field towers and improved optimal rate list decoding , 2012, STOC '12.

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

[36]  Swastik Kopparty List-Decoding Multiplicity Codes , 2012, Theory Comput..

[37]  Shubhangi Saraf,et al.  High-rate codes with sublinear-time decoding , 2010, Electron. Colloquium Comput. Complex..

[38]  Atri Rudra,et al.  ℓ2/ℓ2-Foreach Sparse Recovery with Low Risk , 2013, ICALP.

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

[40]  Atri Rudra,et al.  Average-radius list-recoverability of random linear codes , 2018, SODA.

[41]  Ronitt Rubinfeld,et al.  Robust Characterizations of Polynomials with Applications to Program Testing , 1996, SIAM J. Comput..