The Projective Reed-Muller Code from Quartic Curves in the Projective Plane

We study the Hamming weight enumerator of the projective Reed-Muller code ${\mathcal{C}}_{2,4}$ over ${\mathbf{F}}_q$, with $\mathrm{char}({\mathbf{F}}_q)>3$. We determine the geometrically reducible and singular contributions by combining incidence counts with counts of plane linear series on a fixed normalization. The line and conic configurations and the genus zero cases give explicit polynomials. The line-cubic and genus one cases give formulas involving class numbers. The genus two contribution is a finite sum over hyperelliptic models, using their point counts over ${\mathbf{F}}_q$ and ${\mathbf{F}}_{q^2}$ and their rational Weierstrass points. The remaining smooth contribution is the weighted point count distribution of nonhyperelliptic genus three curves.