PreprintCC BY 4.0arXiv
The Projective Reed-Muller Code from Quartic Curves in the Projective Plane
Abstract
We study the Hamming weight enumerator of the projective Reed-Muller code π2,4 over Fq, with char(Fq)>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 Fq and Fq2 and their rational Weierstrass points. The remaining smooth contribution is the weighted point count distribution of nonhyperelliptic genus three curves.
Related
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- On the Bourbaki Degree of Plane Projective Curves
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BibTeX
@misc{zhuang2026projective,
title = {{The Projective Reed-Muller Code from Quartic Curves in the Projective Plane}},
author = {Xu Zhuang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27392},
eprint = {2610.27392},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27392},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Projective Reed-Muller Code from Quartic Curves in the Projective Plane
AU - Xu Zhuang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27392
UR - https://arxiv.org/abs/2610.27392
AB - We study the Hamming weight enumerator of the projective Reed-Muller code π_2,4 over F_q, with char(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 F_q and F_q^2 and their rational Weierstrass points. The remaining smooth contribution is the weighted point count distribution of nonhyperelliptic genus three curves.
ER -