Calculation of Electroproduction to Nnlo and Precision Determination of Α S

We use the known values of the two loop Wilson coefficients and the three loop anomalous dimension matrix γ(n) to perform a next-to-next-to leading order (NNLO) calculation of ep deep inelastic scattering. Because γ(n) is only known for a few values of n, the method of average reconstruction has to be used, which leaves 102 effective experimental points, for 12 parameters: the QCD mass Λ, and 11 initial values for the moments of the structure functions. The data points spread in the range of momenta 2.5 GeV 2 ≤ Q 2 ≤ 230 GeV 2. The χ 2 /d.o.f. decreases substantially when going from LO to NLO, and also from NLO to NNLO (although only a little now) to χ 2 /d.o.f. corresponding to the value of the coupling at the Z mass of α (3 loop) s (M 2 Z) = 0.1172 ± 0.0024. The calculation, which constitutes a very precise test of QCD, includes target mass corrections; the error takes into account experimental errors and higher twist effects among other estimated theoretical errors.