Traces of Hecke Operators and Refined Weight Enumerators of Reed-Solomon Codes

TL;DR

This paper derives formulas for the quadratic residue weight enumerators of certain Reed-Solomon codes, linking their coefficients to traces of Hecke operators on modular forms, using the Eichler-Selberg trace formula.

Contribution

It provides explicit formulas for weight enumerator coefficients of dual projective Reed-Solomon codes involving Hecke operator traces, connecting coding theory with modular forms.

Findings

Formulas for weight enumerator coefficients involving Hecke traces

Connection between Reed-Solomon codes and modular forms

Application of Eichler-Selberg trace formula to coding theory

Abstract

We study the quadratic residue weight enumerators of the dual projective Reed-Solomon codes of dimensions $5$ and $q-4$ over the finite field $\mathbb{F}_q$. Our main results are formulas for the coefficients of the the quadratic residue weight enumerators for such codes. If $q=p^v$ and we fix $v$ and vary $p$ then our formulas for the coefficients of the dimension $q-4$ code involve only polynomials in $p$ and the trace of the $q$th and $(q/p^2)$th Hecke operators acting on spaces of cusp forms for the congruence groups $\operatorname{SL}_2 (\mathbb{Z}), \Gamma_0(2)$, and $\Gamma_0(4)$. The main tool we use is the Eichler-Selberg trace formula, which gives along the way a variation of a theorem of Birch on the distribution of rational point counts for elliptic curves with prescribed $2$-torsion over a fixed finite field.