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We present the scheme-invariant unpolarized and polarized flavor non-singletevolution equation to N$^3$LO for the structure functions $F_2(x,Q^2)$ and$g_1(x,Q^2)$ including the charm- and bottom quark effects in the asymptoticrepresentation. The corresponding evolution is based on the experimentalmeasurement of the non-singlet structure functions at a starting scale $Q_0^2$.In this way the evolution does only depend on the strong coupling constant$\alpha_s(M_Z)$ or the QCD scale $\Lambda_{\rm QCD}$ and the charm and bottomquark masses $m_c$ and $m_b$ and provides one of the cleanest ways to measurethe strong coupling constant in future high luminosity deep-inelasticscattering experiments. The yet unknown parts of the 4-loop anomalousdimensions introduce only a marginal error in this analysis.

The N$^3$LO Scheme-invariant QCD Evolution of the Non-singlet Structure Functions \boldmath $F^{\rm NS}_2(x,Q^2)$ and $g_1^{\rm NS}(x,Q^2)$