Weighted Reed-Muller codes revisited

TL;DR

This paper revisits weighted Reed-Muller codes, optimizing weights for two-variable cases, analyzing the impact of point set ratios on minimum distance, and introduces list decoding algorithms that outperform previous error correction capabilities.

Contribution

It determines optimal weights for weighted Reed-Muller codes in two variables and develops improved list decoding algorithms for affine variety codes.

Findings

Weighted Reed-Muller codes outperform previous reputation.

Optimal weights depend on the ratio of point set sizes.

One decoding algorithm corrects up to 31 errors in a specific code.

Abstract

We consider weighted Reed-Muller codes over point ensemble $S_1 \times...\times S_m$ 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.