A class of exceptional polynomials

We present a class of indecomposable polynomials of non primepower degree over the finite field of two elements which are permutation polynomials on infinitely many finite extensions of the field. The associated geometric monodromy groups are the simple groups where k ≥ 3 and odd. (The first member of this class was previously found by P. Müller [17]. This realises one of only two possibilities for such a class which remain following deep work of Fried, Guralnick and Saxl [7]. The other is associated with, psl(3) k ≥ 3, and odd in fields of characteristic 3