Reed–Muller codes evaluate polynomials at all points of a finite projective space. Every point has weight one — it counts equally. Weighted projective spaces break this democracy: different coordinates carry different weights, and the geometry deforms accordingly.
The paper on weighted projective Reed–Muller codes (arXiv: 2603.24397) provides a structural analysis of codes defined by evaluating polynomials on weighted projective spaces.
The recursive construction is the key technique. Under specific weight conditions, the code decomposes into pieces that relate to lower-dimensional or lower-degree codes. This recursion is not just computational convenience — it reveals structure. Generalized Hamming weights, which measure the minimum support of subcodes (not just codewords), inherit bounds from the recursion. Subfield subcodes — restrictions to smaller alphabets — and dual codes also admit recursive descriptions.
When the degree is low relative to the weights, the dual code is itself an evaluation code. This duality is structural: the code and its dual both come from polynomial evaluation, just at different degree ranges. For non-degenerate cases, Schur products (componentwise multiplication of codewords) give further structural information.
The through-claim: weighting the space deforms the code but preserves the recursive skeleton. Ordinary projective Reed–Muller codes have clean recursive structure. The weights complicate the geometry but do not destroy the recursion — they modify it. The structural properties (dual codes, subfield subcodes, Hamming weights) all have weighted analogues that track back to the unweighted theory through the recursion.
2603.24397. Coding theory / Reed–Muller codes / weighted projective spaces / Hamming weights / evaluation codes.