Traces of Hecke operators and refined weight enumerators of Reed-Solomon codes

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 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 $q^{\mathrm {th}}$ and $(q/p^2)^{\text {th}}$ 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.