Weight enumerators of Reed-Muller codes from cubic curves and their duals

TL;DR

This paper computes the weight enumerators of certain Reed-Muller codes of order 3 over finite fields, linking algebraic geometry, coding theory, and modular forms, and provides formulas for their duals' weight enumerators.

Contribution

It derives explicit formulas for the weight enumerators of projective and affine Reed-Muller codes of order 3, connecting them to cubic curves and Hecke operators.

Findings

Weight enumerators of specific Reed-Muller codes are explicitly computed.

Formulas for dual codes' weight enumerators involve traces of Hecke operators.

Connections between coding theory, algebraic geometry, and modular forms are established.

Abstract

Let $\mathbb{F}_q$ be a finite field of characteristic not equal to $2$ or $3$. We compute the weight enumerators of some projective and affine Reed-Muller codes of order $3$ over $\mathbb{F}_q$. These weight enumerators answer enumerative questions about plane cubic curves. We apply the MacWilliams theorem to give formulas for coefficients of the weight enumerator of the duals of these codes. We see how traces of Hecke operators acting on spaces of cusp forms for $\operatorname{SL}_2(\mathbb{Z})$ play a role in these formulas.