  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.
