Algebraic Invariants of Codes on Weighted Projective Planes

TL;DR

This paper investigates the algebraic structure of weighted projective Reed-Muller codes on spaces like P(1,1,a), using advanced algebraic techniques to compute key code parameters and invariants.

Contribution

It introduces a novel algebraic approach to analyze weighted projective Reed-Muller codes, including computing minimal free resolutions and Hilbert series.

Findings

Computed minimal free resolutions for the codes.

Determined Hilbert series of the vanishing ideals.

Calculated main parameters of the codes.

Abstract

Weighted projective spaces are natural generalizations of projective spaces with a rich structure. Projective Reed-Muller codes are error-correcting codes that played an important role in reliably transmitting information on digital communication channels. In this case study, we explore the power of commutative and homological algebraic techniques to study weighted projective Reed-Muller (WPRM) codes on weighted projective spaces of the form $\mathbb{P}(1,1,a)$. We compute minimal free resolutions and thereby obtain Hilbert series for the vanishing ideal of the $\mathbb{F}_q$-rational points, and compute main parameters for these codes.