Guest Column: Local Testing and Decoding of High-Rate Error-Correcting Codes

We survey the state of the art in constructions of locally testable codes, locally decodable codes and locally correctable codes of high rate.

[1]  Oded Goldreich,et al.  Bravely, Moderately: A Common Theme in Four Recent Works , 2011, Studies in Complexity and Cryptography.

[2]  KoppartySwastik,et al.  Constant Rate PCPs for Circuit-SAT with Sublinear Query Complexity , 2016 .

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

[4]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

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

[6]  Or Meir,et al.  The tensor product of two good codes is not necessarily robustly testable , 2012, Inf. Process. Lett..

[7]  Rafail Ostrovsky,et al.  Local Correctability of Expander Codes , 2013, ICALP.

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

[9]  Michael Viderman A Combination of Testability and Decodability by Tensor Products , 2012, APPROX-RANDOM.

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

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

[12]  Carsten Lund,et al.  Algebraic methods for interactive proof systems , 1992, JACM.

[13]  Ran Raz,et al.  A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP , 1997, STOC '97.

[14]  Avi Wigderson,et al.  Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[15]  Sergey Yekhanin,et al.  Towards 3-query locally decodable codes of subexponential length , 2008, JACM.

[16]  Carsten Lund,et al.  Proof verification and the intractability of approximation problems , 1992, FOCS 1992.

[17]  Atri Rudra,et al.  On the Robust Testability of Product of Codes , 2005, Electron. Colloquium Comput. Complex..

[18]  Avi Wigderson,et al.  Robust local testability of tensor products of LDPC codes ? , 2006 .

[19]  Madhu Sudan,et al.  Some improvements to total degree tests , 1995, Proceedings Third Israel Symposium on the Theory of Computing and Systems.

[20]  Omer Reingold,et al.  Undirected connectivity in log-space , 2008, JACM.

[21]  Eli Ben-Sasson,et al.  Short PCPs with Polylog Query Complexity , 2008, SIAM J. Comput..

[22]  Or Meir,et al.  Combinatorial construction of locally testable codes , 2008, SIAM J. Comput..

[23]  Shubhangi Saraf,et al.  High-rate codes with sublinear-time decoding , 2014, Electron. Colloquium Comput. Complex..

[24]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[25]  Eyal Kushilevitz,et al.  Private information retrieval , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[26]  Or Meir,et al.  Constant Rate PCPs for Circuit-SAT with Sublinear Query Complexity , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[27]  Robust locally testable codes and products of codes , 2006 .

[28]  Alan Guo,et al.  New affine-invariant codes from lifting , 2012, ITCS '13.

[29]  Klim Efremenko,et al.  3-Query Locally Decodable Codes of Subexponential Length , 2008 .

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

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

[32]  Noga Alon Combinatorial Nullstellensatz , 1999, Combinatorics, Probability and Computing.

[33]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

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

[35]  Zeev Dvir,et al.  Matching Vector Codes , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[36]  Swastik Kopparty Some remarks on multiplicity codes , 2013, Discrete Geometry and Algebraic Combinatorics.

[37]  Alan Guo,et al.  High-Rate Locally Correctable Codes via Lifting , 2013, IEEE Transactions on Information Theory.

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

[39]  Eli Ben-Sasson,et al.  Tensor Products of Weakly Smooth Codes are Robust , 2008, Theory Comput..

[40]  Madhu Sudan,et al.  Algebraic property testing: the role of invariance , 2008, STOC '08.

[41]  Paul Valiant,et al.  The Tensor Product of Two Codes Is Not Necessarily Robustly Testable , 2005, APPROX-RANDOM.