List Decoding of q-ary Reed-Muller Codes

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Author(s)
Wu, Xin-Wen
Griffith University Author(s)
Year published
2004
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The q-ary Reed-Muller (RM) codes RMq(u,m) of length n=qm 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 RMq(u,m) with uqm. Then, using the list- decoding algorithm in Guruswami and Sudan (1999) for RS codes over Fqm, we present a list-decoding algorithm for q-ary RM codes. This algorithm is applicable ...
View more >The q-ary Reed-Muller (RM) codes RMq(u,m) of length n=qm 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 RMq(u,m) with uqm. Then, using the list- decoding algorithm in Guruswami and Sudan (1999) for RS codes over Fqm, 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-√(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 Fq and O(n3) field operations in Fqm under some assumption.
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View more >The q-ary Reed-Muller (RM) codes RMq(u,m) of length n=qm 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 RMq(u,m) with uqm. Then, using the list- decoding algorithm in Guruswami and Sudan (1999) for RS codes over Fqm, 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-√(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 Fq and O(n3) field operations in Fqm under some assumption.
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Journal Title
IEEE Transactions on Information Theory
Volume
50
Issue
4
Copyright Statement
© 2004 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
Subject
Coding and Information Theory
Artificial Intelligence and Image Processing
Electrical and Electronic Engineering
Communications Technologies