On maximality of a class of binomial Artin--Schreier curves in characteristic two

We study a class of binomial Artin--Schreier curves in characteristic two, generalizing Suzuki curves. We show that they are maximal under a certain condition. If the condition is not satisfied, we show that they are non-maximal over any field extension. In special cases, we explicitly calculate the number of rational points and the $L$-polynomials of the curves.