Far from a point in the \(F_4(q)\) geometry (Q5927682)

The \(F_4(q)\) geometry of the title is precisely the point-line geometry \(F_{4,1}(q)\) arising from the building of type \(F_4\) over the finite field with \(q\) elements and of absolute type \(F_4\) (split case), by taking as points the elements of type 1 and as lines the elements of type 2 (Bourbaki numbering). Fixing a point \(x\) of that geometry, the author investigates the subgeometry induced by the points at maximal distance from \(x\). The main result is the determination of all parameters of a 12-class association scheme on the \(q^{15}\) points of that subgeometry in the case where \(q\) is even. This is obtained by a careful combination of geometric, combinatorial and group theoretic arguments (the association scheme joins certain classes of a groups scheme for the stabilizer of \(x\) in \(\Aut(F_{4,1} (q))\).NEWLINENEWLINENEWLINETo me it is not so much the result, but rather the way it is obtained and proved that makes the paper interesting and of high quality. In fact, the case \(q=2\) was done before in the PhD thesis of Remko Riebeek (1998) with the help of a computer. I particularly like it that the author does a much more general case by hand.