Elation KM-arcs (Q2322542)

A KM-arc of type \(t\) in the desarguesian projective plane of order \(q\) is a set of \(q+t\) points such that every line meets the set of points in \(0\), \(2\) or \(q+t\) points. If \(A\) is a KM-arc of type \(t>2\) with \(t\)-nucleus \(N\), then \(A\) is an elation KM-arc with elation line \(\ell_\infty\) if and only if for every \(t\)-secant \(\ell \neq \ell_\infty\), the group of elations with axis \(\ell_\infty\) that stabilize the arc acts transitively on the points of \(\ell\). The authors give an algebraic framework for elation KM-arcs and they prove that all elation KM-arcs of type \(\frac{q}{4}\) in the desarguesian plane are translation KM-arcs. Using a previous result of the authors, this new result completes the classification problem for KM-arcs of this type. Additionally, they construct an infinite family of elation KM-arcs of type \(\frac{q}{8}\) for all \(q=2^h, h>3\) and an infinite family of type \(\frac{q}{16}\) for \(q=2^h\) where \(4,6,7 \vert h\).