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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