Search

We give an algorithmic generalisation of Dickson's method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson's idea is to formulate from Hermite's criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most 6 were essentially deduced from manual analysis of these polynomial equations, but this approach is no longer viable for $d>6$. Our idea is to calculate some radicals of ideals generated by the polynomials, implemented by a computer algebra system. Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree 8 over an arbitrary finite field of odd order $q>8$. Such PPs exist if and only if $q\in \{11,13,19,23,27,29,31\}$ and are explicitly listed in normalised form. [ABSTRACT FROM AUTHOR]

Up to linear transformations, a classification of all permutation polynomials (PPs) of degree 7 over F q 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 F q 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. [ABSTRACT FROM AUTHOR]

Up to linear transformations, we obtain a classification of permutation polynomials (PPs) of degree 8 over F 2 r with r > 3. By Bartoli et al. (2017) [1] , a polynomial f of degree 8 over F 2 r is exceptional if and only if f − f (0) is a linearized PP, which has already been classified. So it suffices to search for non-exceptional PPs of degree 8 over F 2 r , which exist only when r ⩽ 9 by a 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 explicit equations by Hermite's criterion are provided for the polynomial coefficients. The main result is that a non-exceptional PP f of degree 8 over F 2 r (with r > 3) exists if and only if r ∈ { 4 , 5 , 6 } , and such f is explicitly listed up to linear transformations. [ABSTRACT FROM AUTHOR]