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

TL;DR

This paper introduces a simplified formula for van der Geer--van der Vlugt curves in characteristic 2 using additive polynomial factorization, enabling explicit computations and classifications of these curves.

Contribution

It presents a new, simpler formula for Frobenius eigenvalues of these curves, facilitating their explicit construction and analysis.

Findings

Derived a new explicit formula for Frobenius eigenvalues

Provided a method to construct all maximal and minimal curves of this type

Computed examples and studied periods of these curves

Abstract

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.