  We produce a new family of polynomials f(x) over fields K of characteristic 2
which are exceptional, in the sense that f(x)-f(y) has no absolutely
irreducible factors in K[x,y] besides the scalar multiples of x-y; when K is
finite, this condition is equivalent to saying there are infinitely many finite
extensions L/K for which the map c --> f(c) is bijective on L. Our polynomials
have degree (2^e-1)*2^(e-1), where e is odd. Combined with our previous paper
arxiv:0707.1835, this completes the classification of indecomposable
exceptional polynomials of degree not a power of the characteristic. The
strategy of our proof is to identify the curves that can arise as the Galois
closure of the branched cover P^1 --> P^1 induced by an exceptional polynomial
f. In this case, the curves turn out to be x^(q+1)+y^(q+1)=a+T(xy), where
T(z)=z^(q/2)+z^(q/4)+...+z. Our proofs rely on new properties of ramification
in Galois covers of curves, as well as the computation of the automorphism
groups of all curves in a certain 2-parameter family.
