List decoding of q-ary Reed-Muller codes

The q-ary Reed-Muller (RM) codes RM/sub q/(u,m) of length n=q/sup m/ are a generalization of Reed-Solomon (RS) codes, which use polynomials in m variables to encode messages through functional encoding. Using an idea of reducing the multivariate case to the univariate case, randomized list-decoding algorithms for RM codes were given in and . The algorithm in Sudan et al. (1999) is an improvement of the algorithm in , it is applicable to codes RM/sub q/(u,m) with u<q/2 and works for up to E<n(1-/spl radic/2u/q) errors. In this correspondence, following , we show that q-ary RM codes are subfield subcodes of RS codes over F/sub q//sup m/. Then, using the list- decoding algorithm in Guruswami and Sudan (1999) for RS codes over F/sub q//sup m/, we present a list-decoding algorithm for q-ary RM codes. This algorithm is applicable to codes of any rates, and achieves an error-correction bound n(1-/spl radic/(n-d)/n). The algorithm achieves a better error-correction bound than the algorithm in , since when u is small. The implementation of the algorithm requires O(n) field operations in F/sub q/ and O(n/sup 3/) field operations in F/sub q//sup m/ under some assumption.

[1]  Amin Shokrollahi,et al.  List Decoding of Algebraic-Geometric Codes , 1999, IEEE Trans. Inf. Theory.

[2]  Xin-Wen Wu,et al.  Efficient root-finding algorithm with application to list decoding of Algebraic-Geometric codes , 2001, IEEE Trans. Inf. Theory.

[3]  M. Kerimov.  The theory of error-correcting codes☆ , 1980 .

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

[5]  Joan Borrell,et al.  A fast algorithm to compute irreducible and primitive polynomials in finite fields , 2005, Mathematical systems theory.

[6]  Victor Shoup,et al.  Fast construction of irreducible polynomials over finite fields , 1994, SODA '93.

[7]  B. Lindström,et al.  A Generalization of a Combinatorial Theorem of Macaulay , 1969 .

[8]  Tom Høholdt,et al.  Footprints or generalized Bezout's theorem , 2000, IEEE Trans. Inf. Theory.

[9]  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).

[10]  Ruud Pellikaan,et al.  Which linear codes are algebraic-geometric? , 1991, IEEE Trans. Inf. Theory.

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

[12]  Victor K.-W. Wei,et al.  Generalized Hamming weights for linear codes , 1991, IEEE Trans. Inf. Theory.

[13]  Tom Høholdt,et al.  Decoding Hermitian Codes with Sudan's Algorithm , 1999, AAECC.

[14]  Ron M. Roth,et al.  Efficient decoding of Reed-Solomon codes beyond half the minimum distance , 2000, IEEE Trans. Inf. Theory.

[15]  Xin-Wen Wu.  An algorithm for finding the roots of the polynomials over order domains , 2002, Proceedings IEEE International Symposium on Information Theory,.

[16]  Vladimir M. Blinovsky,et al.  List decoding , 1992, Discret. Math..

[17]  Ruud Pellikaan,et al.  Decoding geometric Goppa codes using an extra place , 1992, IEEE Trans. Inf. Theory.

[18]  Philippe Delsarte,et al.  On subfield subcodes of modified Reed-Solomon codes (Corresp.) , 1975, IEEE Trans. Inf. Theory.

[19]  Michael Ben-Or,et al.  Probabilistic algorithms in finite fields , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[20]  Ruud Pellikaan,et al.  Generalized Hamming Weights of q-ary Reed-Muller Codes , 1998, IEEE Trans. Inf. Theory.

[21]  Ronitt Rubinfeld,et al.  Learning polynomials with queries: The highly noisy case , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[22]  Peter Elias,et al.  List decoding for noisy channels , 1957 .

[23]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

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

[25]  F. S. Macaulay.  Some Properties of Enumeration in the Theory of Modular Systems , 1927 .

[26]  Igor E. Shparlinski,et al.  Finding irreducible and primitive polynomials , 1993, Applicable Algebra in Engineering, Communication and Computing.

[27]  Tadao Kasami,et al.  New generalizations of the Reed-Muller codes-I: Primitive codes , 1968, IEEE Trans. Inf. Theory.

[28]  Neil J. A. Sloane,et al.  The theory of error-correcting codes (north-holland , 1977 .

[29]  Igor E. Shparlinski.  Finite Fields: Theory and Computation , 1999 .

[30]  Ruud Pellikaan,et al.  On the Structure of Order Domains , 2002 .