Factorization of Additive Polynomials and van der Geer--van der Vlugt curves in characteristic 2

In our previous work, we gave a formula for the Frobenius eigenvalues of van der Geer–van der Vlugt curves in characteristic 2 by considering suitable quotients of the curve. Although the formula is explicit, it depends on many choices, which makes the formula complicated. In this article, we take a different approach using a factorization of additive polynomials, and prove a new formula. The resulting formula is simpler and is useful for explicit computations. As applications, we provide a method for constructing maximal and minimal van der Geer–van der Vlugt curves, and show that every such curve arises from this construction. We also compute various examples of van der Geer–van der Vlugt curves and study their periods.