Weighted Reed-Muller codes revisited

Hans Olav Geil; Casper Thomsen

doi:10.1007/s10623-012-9680-8

Weighted Reed-Muller codes revisited

Hans Olav Geil, Casper Thomsen

Department of Mathematical Sciences

Research output: Contribution to journal › Journal article › Research › peer-review

33 Citations (Scopus)

Abstract

We consider weighted Reed–Muller codes over point ensemble S₁ × · · · × S_m
where S_i needs not be of the same size as S_j. For m = 2 we determine optimal weights and analyze in detail what is the impact of the ratio |S₁|/|S₂| on the minimum distance. In conclusion the weighted Reed–Muller code construction is much better than its reputation.
For a class of affine variety codes that contains the weighted Reed–Muller codes we then present two list decoding algorithms. With a small modification one of these algorithms is able to correct up to 31 errors of the [49,11,28] Joyner code.

Original language	English
Journal	Designs, Codes and Cryptography
Volume	66
Issue number	1-3
Pages (from-to)	195-220
Number of pages	26
ISSN	0925-1022
DOIs	https://doi.org/10.1007/s10623-012-9680-8
Publication status	Published - 2013

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title = "Weighted Reed-Muller codes revisited",

abstract = "We consider weighted Reed–Muller codes over point ensemble S1 × · · · × Sm where Si needs not be of the same size as Sj. For m = 2 we determine optimal weights and analyze in detail what is the impact of the ratio |S1|/|S2| on the minimum distance. In conclusion the weighted Reed–Muller code construction is much better than its reputation. For a class of affine variety codes that contains the weighted Reed–Muller codes we then present two list decoding algorithms. With a small modification one of these algorithms is able to correct up to 31 errors of the [49,11,28] Joyner code.",

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AU - Thomsen, Casper

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AB - We consider weighted Reed–Muller codes over point ensemble S1 × · · · × Sm where Si needs not be of the same size as Sj. For m = 2 we determine optimal weights and analyze in detail what is the impact of the ratio |S1|/|S2| on the minimum distance. In conclusion the weighted Reed–Muller code construction is much better than its reputation. For a class of affine variety codes that contains the weighted Reed–Muller codes we then present two list decoding algorithms. With a small modification one of these algorithms is able to correct up to 31 errors of the [49,11,28] Joyner code.

U2 - 10.1007/s10623-012-9680-8

DO - 10.1007/s10623-012-9680-8

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JO - Designs, Codes and Cryptography

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Weighted Reed-Muller codes revisited

Abstract

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