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

Introduction

The Hamming weight enumerator of a linear code $C$ of length $\ell$ over $\mathbb{F}_q$ is

where $\operatorname{wt}(c)$ is the number of nonzero coordinates of $c$. The projective Reed–Muller code $C_{m,d}$ is obtained by evaluating the homogeneous degree $d$ forms in $m+1$ variables at chosen vector representatives of the points of $\mathbb{P}^m(\mathbb{F}_q)$. Changing these representatives gives a monomially equivalent code and preserves its weight enumerator [15, 22, 11]. The zero coordinates of the codeword defined by a form $F$ are exactly the points of $V(F)(\mathbb{F}_q)$. In the injective range, computing the weight enumerator therefore amounts to counting equations with each possible number of rational zeros.

We consider ternary quartic forms. Throughout the paper, let $q=p^r$ with $p>3$, fix $\mathbb{F}_q$, and write $\operatorname{Frob}_q$ for the $q$ power Frobenius. Since $q\geq5$, projective evaluation in degree four is injective. Thus $C_{2,4}$ has length $q^2+q+1$, dimension $15$, and weight enumerator

Here $V(0)=\mathbb{P}^2$. Nonzero scalar multiples of a form are distinct codewords, although they define the same plane curve.

We call a form reducible if it factors over $\overline{\mathbb{F}}_q$. For an absolutely irreducible singular quartic, the normalization has genus two, one, or zero. A smooth plane quartic is a nonhyperelliptic genus three curve in its canonical embedding. Let $\mathcal{M}_3^{\mathrm{nhyp}}(\mathbb{F}_q)$ denote representatives of the $\mathbb{F}_q$-isomorphism classes of such smooth curves, and set

The Hasse–Weil bound gives $S_q(n)=0$ unless $\lvert n-(q+1)\rvert\leq6\sqrt{q}$.

The reducible and absolutely irreducible singular contributions are determined by explicit polynomials, formulas involving class numbers, and finite sums over hyperelliptic models of genus two curves, as summarized below. The smooth contribution is expressed in terms of $S_q(n)$, which remains unevaluated for general $q$.

Contribution	Arithmetic data	Form of the answer
Line and conic configurations	Finite plane incidence data	Explicit polynomials
Line–cubic and genus one cases	Elliptic traces and rational two-torsion	Formulas involving class numbers
Genus zero normalization	Frobenius action on singular fibers	Explicit polynomials
Genus two normalization	Point counts over $\mathbb{F}_q$, $\mathbb{F}_{q^2}$ and rational Weierstrass points	Finite hyperelliptic model sums
Smooth quartics	Nonhyperelliptic genus three point distribution	Weighted distribution $S_q(n)$

Theorem 4.1 gives the expanded formula and identifies each coefficient. The factor $(q-1)\#\operatorname{PGL}_3(\mathbb{F}_q)$ in the smooth term accounts for scalar equations and projective coordinates, with automorphisms accounted for in $S_q(n)$.

For absolutely irreducible singular quartics, the common method is to fix the normalization $C$ and count base point free degree four plane linear series that give birational maps onto their images. For each singularity profile, Proposition 3.3 converts this count into a count of equations by multiplying by $(q-1)\#\operatorname{PGL}_3(\mathbb{F}_q)/\#\operatorname{Aut}_{\mathbb{F}_q}(C)$. The Frobenius action on the singular fibers determines the change in rational point count from $C$ to its plane model. Thus the geometric count and its contribution to the enumerator are treated in the same way in each genus.

Two parameter spaces make this method particularly concrete. For a genus two curve $C$, let $\operatorname{Weier}(C)(\mathbb{F}_q)$ denote its rational Weierstrass points, the rational ramification points of the hyperelliptic map. The plane series on $C$ are parametrized by $(\operatorname{Sym}^2 C)(\mathbb{F}_q)\setminus\lvert K_C\rvert(\mathbb{F}_q)$. The excluded canonical divisors give double covers of conics, while the remaining divisors are the singular fibers of quartic models. Proposition 3.4 expresses the split node, cusp, and nonsplit node counts using only $\#C(\mathbb{F}_q)$, $\#C(\mathbb{F}_{q^2})$, and $\#\operatorname{Weier}(C)(\mathbb{F}_q)$. For genus one, projections of an elliptic normal quartic give the parameter space. Proposition 3.5 relates the projection centers to degree two divisors and divisor pairs, identifies the exceptional conic maps, and determines the singular fibers. Counting these data gives the trace shift formulas of Proposition 3.7.

The divisor constructions already occur in Bergström’s classification and count of singular quartics for a sieve used to study the cohomology of genus three moduli spaces. His genus one criterion identifies the degree two maps by the linear equivalence of the two degree two divisors [2 Lemma 13.7]. We use his genus zero counts for two and three geometric singular points [2 Sections 12.3 and 13.4]. For one geometric singular point we give direct derivations using linear series and intrinsic branch data. We also give direct incidence and projection counts for the reducible and genus one cases. We organize these constructions and counts by fixed normalization and trace shift to obtain the coefficients required by this weight enumerator. The line–cubic contribution and the arithmetic evaluation of the elliptic sums use the formulas of Kaplan and Kaplan–Petrow [11, 13].

This continues earlier work on exact weight enumerators of projective Reed–Muller codes. Elkies computed the weight enumerators for quadrics in projective spaces of arbitrary dimension and for cubics in $\mathbb{P}^2$ and $\mathbb{P}^3$ [8]. Aubry obtained related results for projective quadrics [1]. Kaplan studied the weight enumerators of the projective and affine Reed–Muller codes obtained from plane cubics [11]. Kaplan–Matei used the second weight enumerator to count pairs of plane cubics with prescribed rational intersections [12].

The smooth contribution also connects this problem with the study of rational points on genus three curves. Lauter proved that over every finite field there is a genus three curve whose number of rational points lies within three of the Serre–Weil upper or lower bound [16]. The alternative between the two bounds is related to Serre’s obstruction. A principally polarized abelian threefold that is geometrically a Jacobian may require a quadratic twist to become a Jacobian over $\mathbb{F}_q$, and this twist changes the sign of its Frobenius trace. Ritzenthaler computed this obstruction explicitly and applied it to the existence of optimal genus three curves [20].

This obstruction also appears in the distribution of rational point counts. Quadratic twisting pairs elliptic curves, and likewise hyperelliptic curves, with opposite Frobenius traces. For smooth plane quartics, the quadratic twist of the Jacobian need not be a Jacobian over $\mathbb{F}_q$, so there is no analogous pairing. Lercier, Ritzenthaler, Rovetta, Sijsling, and Smith gave a heuristic explanation of the observed asymmetry in the trace distribution and its relation to Serre’s obstruction [17]. More recently, Bergström, Howe, Lorenzo García, and Ritzenthaler refined Katz–Sarnak theory by determining limits of moments that measure this asymmetry for nonhyperelliptic curves of genus at least three [4]. Our formulas isolate the distribution $S_q(n)$ from the reducible and singular contributions, making explicit which part of the quartic enumerator depends on these questions about smooth curves.

Section 2 counts the reducible quartics by their geometric components and Frobenius action. The incidence tables are collected in the appendix. Section 3 applies fixed normalization counting in genus two, one, and zero. Section 4 assembles the enumerator, evaluates the elliptic sums by class numbers, and writes the genus two contribution as a finite sum over all squarefree polynomials of degree five or six. SageMath code and saved outputs are maintained in a companion repository.¹

Reducible quartic forms

We count nonzero quartic forms that factor over $\overline{\mathbb{F}}_q$. The point count depends only on the reduced union of the components. For the line and conic configurations, we count equations up to multiplication by a nonzero scalar using incidence arguments. We treat the line and cubic case using the affine cubic weight enumerator. The full incidence tables are in Section A. We describe the counting methods below.

Line and conic configurations

For quartics with four line factors, the configurations and their counts are listed in Table A.1. We check the total independently by choosing Frobenius orbits of lines, allowing repetitions. Let $n_d$ be the number of such orbits of length $d$. There are $q^{2d}+q^d+1$ lines over $\mathbb{F}_{q^d}$, since lines are points of the dual plane. Removing the lines defined over proper subfields and dividing by $d$ gives $n_1=q^2+q+1$, $n_2=(q^4-q)/2$, $n_3=(q^6+q^3-q^2-q)/3$, and $n_4=(q^8-q^2)/4$. The orbit lengths, counted with repetition, form one of the partitions $1+1+1+1$, $2+1+1$, $2+2$, $3+1$, or $4$. These give the five terms in the total

This agrees with the sum of all 21 rows of Table A.1.

There are $q^5-q^2$ smooth conics over $\mathbb{F}_q$, each with $q+1$ rational points. A fixed smooth conic has $q+1$ rational tangent lines, $q(q+1)/2$ rational lines meeting it in two rational points, and $q(q-1)/2$ rational lines meeting it in no rational point. These counts give Table A.2 directly. For two distinct rational lines, choose an unordered pair of the required types and separate the pairs whose intersection lies on the conic. Inclusion and exclusion then gives Table A.3.

A pair of nonrational conjugate lines has just one rational point, their intersection. Each rational point lies on $(q^2-q)/2$ such pairs. For a fixed smooth conic, choosing this point on or off the conic gives Table A.5.

For two distinct smooth conics, we need only the number $r$ of distinct rational intersection points. Bézout gives $0\le r\le4$. A tangency contributes one point to $r$. If both conics are defined over $\mathbb{F}_q$, their union has $2q+2-r$ rational points. If they are conjugate over $\mathbb{F}_{q^2}$, a rational point on either conic also lies on its conjugate, so their union has exactly $r$ rational points.

We count the two families by the same procedure: choose some rational intersection points, count pairs through them, and remove pairs with more intersection points. For $0\le i\le4$, let $a_i$ be the number of subsets of $i$ points in $\mathbb{P}^2(\mathbb{F}_q)$ with no three collinear, and let $c_i(q)$ be the number of smooth conics through any one such subset. The elementary counts are