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

摘要

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^{extup {th}}$ and $ (q/p^2)^{ext {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.