List decoding

About: List decoding is a research topic. Over the lifetime, 7251 publications have been published within this topic receiving 151182 citations.

Decoding via cross-entropy minimization

Abstract: An intuitive algorithm by Lodge et al. [1992] for iterative decoding of block codes is shown to follow from entropy optimization principles. This approach provides a novel and effective algorithm for the soft-decoding of block codes which have a product structure. >

Minimal permutation sets for decoding the binary Golay codes (Corresp.)

Abstract: For permutation decoding of an e error-correcting linear code, a set of permutations which move all error vectors of weight \leq e out of the information places is needed. A method of finding minimal decoding sets is given, along with minimal sets obtained with this method for the binary Golay codes.

"Soft-decision" decoding of Chinese remainder codes

Abstract: Given n relatively prime integers p/sub 1/ , where m/sub i/=m(mod p/sub i/). The soft-decision decoding problem for the Chinese remainder code is given as input a vector of residues r/spl I.oarr/=(r/sub 1/,...,r/sub n/), a vector of weights , and an agreement parameter t. The goal is to find all messages m /spl isin/ M such that the weighted agreement between the encoding of m and r/spl I.oarr/(i.e., /spl Sigma//sub i/ w/sub i/ summed over all i such that r/sub i/=m(mod pi)) is at least t. Here we give a new algorithm for solving the soft-decision problem for the CRT code that works provided the agreement parameter t is sufficiently large. We derive our algorithm by digging deeper into the algebra underlying the error-correcting algorithms and unveiling an "ideal"-theoretic view of decoding. When all weights are equal to 1, we obtain the more commonly studied "list decoding" problem. List decoding algorithms for the Chinese Remainder Code were given recently by O. Goldreich et al. (1999), and improved by D. Boneh. Their algorithms work for t/spl ges//spl radic/(2knlogp/sub n//logp1) and t/spl ges//spl radic/(knlogp/sub n//logp/sub 1/), respectively. We improve upon the algorithms above by using our soft-decision decoding algorithm with a non-trivial choice of weights, solve the list decoding problem provided t/spl ges//spl radic/(k(n+/spl epsi/)), for arbitrarily small /spl epsi//spl ges/0.

Algebraic-geometric codes and multidimensional cyclic codes: a unified theory and algorithms for decoding using Grobner bases

Abstract: It is proved that any algebraic-geometric (AG) code can be expressed as a cross section of an extended multidimensional cyclic code. Both AG codes and multidimensional cyclic codes are described by a unified theory of linear block codes defined over point sets: AG codes are defined over the points of an algebraic curve, and an m-dimensional cyclic code is defined over the points in m-dimensional space. The power of the unified theory is in its description of decoding techniques using Grobner bases. In order to fit an AG code into this theory, a change of coordinates must be applied to the curve over which the code is defined so that the curve is in special position. For curves in special position, all computations can be performed with polynomials and this also makes it possible to use the theory of Grobner bases. Next, a transform is defined for AG codes which generalizes the discrete Fourier transform. The transform is also related to a Grobner basis, and is useful in setting up the decoding problem. In the decoding problem, a key step is finding a Grobner basis for an error locator ideal. For AG codes, multidimensional cyclic codes, and indeed, any cross section of an extended multidimensional cyclic code, Sakata's algorithm can be used to find linear recursion relations which hold on the syndrome array. In this general context, the authors give a self-contained and simplified presentation of Sakata's algorithm, and present a general framework for decoding algorithms for this family of codes, in which the use of Sakata's algorithm is supplemented by a procedure for extending the syndrome array.