Traces of hecke operators and refined weight enumerators of reed-solomon codes

We study the quadratic residue weight enumerators of the dual projective Reed-Solomon codes of dimensions 5 and q − 4 over the finite field Fq. 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/p2 ) th Hecke operators acting on spaces of cusp forms for the congruence groups SL2(Z), Γ0(2), and Γ0(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