A classification of permutation polynomials of degree 7 over finite fields

Abstract

Up to linear transformations, a classification of all permutation polynomials (PPs) of degree 7 over Fq is obtained for any odd prime power q>7. Since all exceptional polynomials of degree 7 are precisely known up to linear transformations, it suffices to work on non-exceptional PPs of degree 7, which exist only when q⩽409 by our previous result. This can be exhausted by the SageMath software running on a personal computer. To facilitate the computation, some requirements after linear transformations and equations by Hermite's criterion are provided for the polynomial coefficients. The main result is that a non-exceptional PP f of degree 7 exists over a finite field Fq of odd order q>7 if and only if q∈{9,11,13,17,19,23,25,27,31,49}, and f is explicitly listed up to linear transformations.