Determine Galois closures, monodromy, and higher ramification groups

Determine the Galois closures, monodromy groups, and higher ramification groups associated with the projective-line covers defined by the explicit reduced APN permutations P_{m,r}.

Background

The paper determines the finite ramification points, ramification indices, different exponents, branch values, and the behavior at infinity for the maps P_{m,r}: \mathbb{P}^1\to\mathbb{P}^1. Specifically, all finite ramification points lie above the branch value 1, while infinity is also a branch point.

The authors explicitly state that the associated Galois closures, monodromy, and higher ramification groups are not determined. Thus, a natural unresolved problem is to obtain the full covering-theoretic and ramification-theoretic structure beyond the local data established in Theorem \ref{thm:B}.