Efficiently List-Decodable Punctured Reed-Muller Codes

The Reed–Muller (RM) code, encoding <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-variate degree-<inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> polynomials over <inline-formula> <tex-math notation="LaTeX">$ \mathbb {F}_{q}$ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">$d < q$ </tex-math></inline-formula>, with its evaluation on <inline-formula> <tex-math notation="LaTeX">$ \mathbb {F}_{q}^{n}$ </tex-math></inline-formula>, has a relative distance <inline-formula> <tex-math notation="LaTeX">$1-d/q$ </tex-math></inline-formula> and can be list decoded from a <inline-formula> <tex-math notation="LaTeX">$1-O(\sqrt {d/q})$ </tex-math></inline-formula> fraction of errors. In this paper, for <inline-formula> <tex-math notation="LaTeX">$d \ll q$ </tex-math></inline-formula>, we give a length-efficient puncturing of such codes, which (almost) retains the distance and list decodability properties of the RM code, but has a much better rate. Specifically, when <inline-formula> <tex-math notation="LaTeX">$q =\Omega ( d^{2}/ \varepsilon ^{2})$ </tex-math></inline-formula>, we give an explicit rate <inline-formula> <tex-math notation="LaTeX">$\Omega \left ({\frac { \varepsilon }{d!}}\right )$ </tex-math></inline-formula> puncturing of RM codes, which have a relative distance at least <inline-formula> <tex-math notation="LaTeX">$(1- \varepsilon )$ </tex-math></inline-formula> and efficient list decoding up to <inline-formula> <tex-math notation="LaTeX">$(1-\sqrt { \varepsilon })$ </tex-math></inline-formula> error fraction. This almost matches the performance of random puncturings, which work with the weaker field size requirement <inline-formula> <tex-math notation="LaTeX">$q= \Omega ( d/ \varepsilon ^{2})$ </tex-math></inline-formula>. We can also improve the field size requirement to the optimal (up to constant factors) <inline-formula> <tex-math notation="LaTeX">$q =\Omega ( d/ \varepsilon )$ </tex-math></inline-formula>, at the expense of a worse list decoding radius of <inline-formula> <tex-math notation="LaTeX">$1- \varepsilon ^{1/3}$ </tex-math></inline-formula> and rate <inline-formula> <tex-math notation="LaTeX">$\Omega \left ({\frac { \varepsilon ^{2}}{d!}}\right )$ </tex-math></inline-formula>. The first of the above tradeoffs is obtained by substituting for the variables functions with carefully chosen pole orders from an algebraic function field; this leads to a puncturing for which the RM code is a subcode of a certain algebraic-geometric code (which is known to be efficiently list decodable). The second tradeoff is obtained by concatenating this construction with a Reed–Solomon-based multiplication friendly pair, and using the list recovery property of algebraic-geometric codes.

[1]  Claus Fieker,et al.  Computing equations of curves with many points , 2013 .

[2]  D. V. Chudnovsky,et al.  Algebraic complexities and algebraic curves over finite fields , 1987 .

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

[4]  H. Niederreiter,et al.  Rational Points on Curves Over Finite Fields: Theory and Applications , 2001 .

[5]  Emanuele Viola,et al.  Pseudorandom Bits for Polynomials , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[6]  J. Pintz,et al.  The Difference Between Consecutive Primes, II , 2001 .

[7]  Victor Shoup,et al.  A computational introduction to number theory and algebra , 2005 .

[8]  Mary Wootters,et al.  On the list decodability of random linear codes with large error rates , 2013, STOC '13.

[9]  J. Pintz,et al.  The Difference Between Consecutive Primes , 1996 .

[10]  Emanuele Viola,et al.  The Sum of D Small-Bias Generators Fools Polynomials of Degree D , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[11]  Nader H. Bshouty,et al.  Testers and their applications , 2014, ITCS.

[12]  Atri Rudra,et al.  Every list-decodable code for high noise has abundant near-optimal rate puncturings , 2013, STOC.

[13]  Ignacio Cascudo,et al.  Asymptotically Good Ideal Linear Secret Sharing with Strong Multiplication over Any Fixed Finite Field , 2009, CRYPTO.

[14]  Venkatesan Guruswami,et al.  Hitting Sets for Low-Degree Polynomials with Optimal Density , 2014, 2014 IEEE 29th Conference on Computational Complexity (CCC).

[15]  Florian Hess,et al.  Computing Riemann-Roch Spaces in Algebraic Function Fields and Related Topics , 2002, J. Symb. Comput..

[16]  Chi-Jen Lu Hitting Set Generators for Sparse Polynomials over Any Finite Fields , 2012, 2012 IEEE 27th Conference on Computational Complexity.

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

[18]  Shachar Lovett,et al.  Unconditional pseudorandom generators for low degree polynomials , 2008, Theory Comput..

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

[20]  I. Shparlinski,et al.  Curves with many points and multiplication in finite fileds , 1992 .

[21]  P. Ebdos,et al.  ON A PROBLEM OF SIDON IN ADDITIVE NUMBER THEORY, AND ON SOME RELATED PROBLEMS , 2002 .

[22]  Venkatesan Guruswami,et al.  Expander-based constructions of efficiently decodable codes , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[23]  H. Stichtenoth,et al.  A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound , 1995 .

[24]  H. Stichtenoth,et al.  On the Asymptotic Behaviour of Some Towers of Function Fields over Finite Fields , 1996 .

[25]  Venkatesan Guruswami,et al.  Optimal rate algebraic list decoding using narrow ray class fields , 2015, J. Comb. Theory, Ser. A.

[26]  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).

[27]  S. G. Vladut,et al.  Algebraic-Geometric Codes , 1991 .

[28]  Andrej Bogdanov Pseudorandom generators for low degree polynomials , 2005, STOC '05.

[29]  Kenneth W. Shum,et al.  A low-complexity algorithm for the construction of algebraic-geometric codes better than the Gilbert-Varshamov bound , 2001, IEEE Trans. Inf. Theory.

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

[31]  Zeev Dvir,et al.  Noisy Interpolating Sets for Low Degree Polynomials , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[32]  Amnon Ta-Shma,et al.  Pseudorandom Generators for Low Degree Polynomials from Algebraic Geometry Codes , 2013, Electron. Colloquium Comput. Complex..