The results of approximate N$^$3LO and detailed NNLO fits to $xF_3$ data of the CCFR97 collaboration are presented. We demonstrate that $1/Q^2$ non-perturbative corrections to $xF_3$ modeled by three independent procedures are shadowed by perturbative QCD effects, starting at the NNLO. Special attention is paid to revealing the role of the recently calculated NNLO corrections to the anomalous dimensions and $N^3$LO corrections to the coefficient functions of odd moments of $xF_3$ with $nleq 13$. The related values of $alpha_s(M_Z)$ are extracted.
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