Codes on Weighted Projective Planes

TL;DR

This paper studies weighted projective Reed-Muller codes on weighted projective planes, providing algebraic and combinatorial tools to determine code parameters like dimension and minimum distance.

Contribution

It introduces a universal Gr"obner basis for the vanishing ideal and a new combinatorial method to analyze the regularity set of rational points.

Findings

Explicit Gr"obner basis for the vanishing ideal

New combinatorial approach to regularity set

Footprint techniques for minimum distance calculation

Abstract

We comprehensively study weighted projective Reed-Muller (WPRM) codes on weighted projective planes $\mathbb{P}(1,a,b)$. We provide the universal Gr\"obner basis for the vanishing ideal of the set $Y$ of $\mathbb{F}_q$--rational points of $\mathbb{P}(1,a,b)$ to get the dimension of the code. We determine the regularity set of $Y$ using a novel combinatorial approach. We employ footprint techniques to compute the minimum distance.