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Weighted Reed–Muller codes revisited

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Abstract

We consider weighted Reed–Muller codes over point ensemble S 1 × · · · × 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 |S 1|/|S 2| 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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  1. Department of Mathematical Sciences, Aalborg University, Aalborg, Denmark

    Olav Geil & Casper Thomsen

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  1. Olav Geil
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  2. Casper Thomsen
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Correspondence to Olav Geil.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Geil, O., Thomsen, C. Weighted Reed–Muller codes revisited. Des. Codes Cryptogr. 66, 195–220 (2013). https://doi.org/10.1007/s10623-012-9680-8

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  • DOI: https://doi.org/10.1007/s10623-012-9680-8

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