Structure of weighted projective Reed-Muller codes

TL;DR

This paper explores the structural properties of weighted projective Reed-Muller codes, providing recursive constructions, bounds on Hamming weights, and insights into their duals and Schur products, advancing understanding of these algebraic codes.

Contribution

It introduces recursive constructions and bounds for weighted projective Reed-Muller codes, and characterizes their duals and Schur products, expanding theoretical understanding.

Findings

Recursive construction for codes under certain weights

Bounds on generalized Hamming weights

Description of dual codes as evaluation codes at low degrees

Abstract

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.