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Permutation polynomials of degree 8 over finite fields of odd characteristic
This paper provides an algorithmic generalization of Dickson's method of
classifying permutation polynomials (PPs) of a given degree 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
. Previous classifications of PPs of degree at most were essentially
deduced from manual analysis of these polynomial equations. However, these
polynomials, needed for that purpose when , are too complicated to solve.
Our idea is to make them more solvable by calculating some radicals of ideals
generated by them, implemented by a computer algebra system (CAS). Our
algorithms running in SageMath 8.6 on a personal computer work very fast to
determine all PPs of degree over an arbitrary finite field of odd order
. The main result is that for an odd prime power , a PP of degree
exists over the finite field of order if and only if
and , and is explicitly listed up to
linear transformations.Comment: 15 page
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