Structure of weighted projective Reed-Muller codes
This work addresses the theoretical understanding of weighted projective Reed-Muller codes, which is incremental as it builds on existing code theory without introducing a new paradigm or broad practical impact.
The paper tackles the structural analysis of weighted projective Reed-Muller codes by providing a recursive construction under certain weight conditions, which is used to derive bounds on generalized Hamming weights and construct subfield subcodes and dual codes, with dual codes further described as evaluation codes for low degrees and insights into Schur products when non-degenerate.
We provide a comprehensive overview of the fundamental structural properties of weighted projective Reed-Muller codes. We give a recursive construction for these codes, under some conditions for the weights, and we use it to derive bounds on the generalized Hamming weights and to obtain a recursive construction for their subfield subcodes and their dual codes. The dual codes are further studied in more generality, where the recursive constructions may not apply, obtaining a description as an evaluation code when the degree is low. We also provide insights into the Schur products of these codes when they are not degenerate.