A Fourier-Analytic Approach to Reed-Muller Decoding

We present a Fourier-analytic approach to list-decoding Reed-Muller codes over arbitrary finite fields. We use this to show that quadratic forms over any field are locally list-decodable up to their minimum distance. The analogous statement for linear polynomials was proved in the celebrated works of Goldreich Previously, tight bounds for quadratic polynomials were known only for q = 2 and 3; the best bound known for other fields was the Johnson radius. Departing from previous work on Reed-Muller decoding which relies on some form of self-corrector, our work applies ideas from Fourier analysis of Boolean functions to low-degree polynomials over finite fields, in conjunction with results about the weight-distribution. We believe that the techniques used here could find other applications, we present some applications to testing and learning.

[1]  Rocco A. Servedio,et al.  Agnostically Learning Halfspaces , 2005, FOCS.

[2]  Oded Regev,et al.  The Learning with Errors Problem (Invited Survey) , 2010, 2010 IEEE 25th Annual Conference on Computational Complexity.

[3]  Ben Green,et al.  The distribution of polynomials over finite fields, with applications to the Gowers norms , 2007, Contributions Discret. Math..

[4]  Nathan Linial,et al.  The influence of variables on Boolean functions , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

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

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

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

[8]  Mihir Bellare,et al.  Linearity testing in characteristic two , 1996, IEEE Trans. Inf. Theory.

[9]  Parikshit Gopalan,et al.  A Fourier-Analytic Approach to Reed–Muller Decoding , 2010, IEEE Transactions on Information Theory.

[10]  Madhu Sudan,et al.  Decodability of group homomorphisms beyond the johnson bound , 2008, STOC '08.

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

[12]  Vitaly Feldman,et al.  New Results for Learning Noisy Parities and Halfspaces , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

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

[14]  Selmer M. Johnson.  A new upper bound for error-correcting codes , 1962, IRE Trans. Inf. Theory.

[15]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

[16]  Edward F. Assmus.  On the Reed-Muller codes , 1992, Discret. Math..

[17]  Ronitt Rubinfeld,et al.  Learning Polynomials with Queries: The Highly Noisy Case , 2000, SIAM J. Discret. Math..

[18]  Selmer M. Johnson.  Improved asymptotic bounds for error-correcting codes , 1963, IEEE Trans. Inf. Theory.

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

[20]  Amnon Ta-Shma,et al.  Extractors from Reed-Muller codes , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[21]  Ryan O'Donnell,et al.  Testing Fourier Dimensionality and Sparsity , 2009, ICALP.

[22]  Elad Haramaty,et al.  On the structure of cubic and quartic polynomials , 2009, STOC '10.

[23]  Shafi Goldwasser,et al.  Proving hard-core predicates using list decoding , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[24]  Prakash Narayan,et al.  Reliable Communication Under Channel Uncertainty , 1998, IEEE Trans. Inf. Theory.

[25]  Atri Rudra,et al.  Testing Low-Degree Polynomials over Prime Fields , 2004, FOCS.

[26]  Prasad Raghavendra,et al.  List decoding tensor products and interleaved codes , 2008, STOC '09.

[27]  Xin-Wen Wu,et al.  List decoding of q-ary Reed-Muller codes , 2004, IEEE Transactions on Information Theory.

[28]  Noga Alon,et al.  Testing Reed-Muller codes , 2005, IEEE Transactions on Information Theory.

[29]  Avi Wigderson,et al.  Simple analysis of graph tests for linearity and PCP , 2003, Random Struct. Algorithms.

[30]  Dana Ron,et al.  Testing polynomials over general fields , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

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

[32]  Shachar Lovett,et al.  Weight Distribution and List-Decoding Size of Reed–Muller Codes , 2012, IEEE Transactions on Information Theory.

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

[34]  David Zuckerman,et al.  List-decoding reed-muller codes over small fields , 2008, STOC '08.

[35]  Moses Charikar,et al.  Local Global Tradeoffs in Metric Embeddings , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[36]  Luca Trevisan,et al.  Pseudorandom generators without the XOR lemma , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

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

[38]  Madhu Sudan,et al.  Improved Low-Degree Testing and its Applications , 2003, Comb..

[39]  Tadao Kasami,et al.  On the weight structure of Reed-Muller codes , 1970, IEEE Trans. Inf. Theory.

[40]  Luca Trevisan,et al.  List-decoding using the XOR lemma , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[41]  Shachar Lovett,et al.  Worst Case to Average Case Reductions for Polynomials , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[42]  Eyal Kushilevitz,et al.  Learning Decision Trees Using the Fourier Spectrum , 1993, SIAM J. Comput..

[43]  Alex Samorodnitsky,et al.  Low-degree tests at large distances , 2006, STOC '07.