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.

翻译：本文全面概述了加权射影Reed-Muller码的基本结构性质。在权重满足特定条件下，我们给出了这些码的递归构造，并利用该构造推导了广义汉明权重的界，同时获得了其子域子码与对偶码的递归构造。对于递归构造可能不适用的一般情形，我们进一步研究了对偶码，在低次数情形下将其描述为求值码。此外，我们还对这些码在非退化情况下的Schur乘积提供了深刻见解。