Weighted Reed-Muller codes revisited

We consider weighted Reed-Muller codes over point ensemble $S_1 \times…\times S_m$where$S_i$needsnotbeofthesamesizeas$S_j$\mathrm{.}For$m = 2$ we determine optimal weights and analyze in detail what is the impact of the ratio $\mathrm{\mid }{S}_1\mathrm{\mid }\mathrm{/}\mathrm{\mid }{S}_2\mathrm{\mid }$ 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.