Permutation polynomials of degree 8 over finite fields of characteristic 2 (Q1987103)

Let \(\mathbb{F}_q\) denote the finite field of characteristic \(p\) and order \(q = p^r\), \(r\) a positive integer. We call \(f \in\mathbb{F}_q[x]\) a permutation polynomial (PP) of \(\mathbb{F}_q\) if the induced map \(a \mapsto f(a)\) permutes \(\mathbb{F}_q\). An exceptional polynomial \(f \in\mathbb{F}_q[x]\) is a permutation polynomial of \(\mathbb{F}_q\) which is PP over infinite many extensions \(\mathbb{F}_{q^m}\) of \(\mathbb{F}_q\). It is known that permutation polynomials of \(\mathbb{F}_q\) of degree \(d<\sqrt[4]{q}\) are in particular exceptional polynomials. Permutation polynomials of small degree have been completely classified: in [\textit{L. E. Dickson}, Ann. Math. 11, 65--120, 161--183 (1896; JFM 28.0135.03)] for \(d\leq 5\) and any \(q\), and for \(d = 6\), \(q\) odd; in [\textit{J. Li} et al., Finite Fields Appl. 16, No. 6, 406--419 (2010; Zbl 1206.11145)] for \(d = 6, 7\), \(q\geq 8\) even; in [\textit{X. Fan}, Finite Fields Appl. 59, 1--21 (2019; Zbl 1444.11238)] for \(d = 7\), \(q\) odd; in [\textit{X. Fan}, Bull. Aust. Math. Soc. 101, No. 1, 40--55 (2020; Zbl 1456.11227)] for \(d=8\), \(q\) odd. In this paper the author classifies permutation polynomials of degree \(8\) over \(\mathbb{F}_{2^r}\), \(r>3\), up to linear transformations. Since the whole set of exceptional polynomials of degree \(8\) over fields of even characteristic have been already determined in [\textit{D. Bartoli} et al., J. Number Theory 176, 46--66 (2017; Zbl 1364.11150)], to complete the classification of PPs of degree \(8\) over \(\mathbb{F}_{2^r}\), \(r>3\), it suffices to search for the non-exceptional ones. In particular, the search can be focused only on the cases \(r\leq 9\). The classification of permutation polynomials has been done with the help of the open-source computer algebra system SageMath. Most of the efforts in this paper are devoted to prune the search space, by proving necessary conditions on the coefficients of a polynomial to be a permutation.