On the uniqueness of a regular thin near octagon on 288 vertices (or the semibiplane belonging to the Mathieu group \(M_{12}\)) (Q1318790)

A partial 2-geometry is a semibiplane with the following additional property: Given an antiflag \((p,B)\), there are precisely \(e\) blocks through \(p\) meeting \(B\) (for some constant \(e\)). The author discusses the case \(e=5\); the possible block sizes \(k\) are then 5, 6, 10, 12, 20. The first three cases are well-known, while the existence question for \(k=20\) is open. For \(k=12\) there is a dual pair of non-isomorphic examples (discovered by Doug Leonard in his Ph.D. thesis). The author shows the uniqueness of these two designs; the proof proceeds via studying associated strongly regular and bipartite distance regular graphs (the ``regular thin near octogons'' in the title). These structures are associated with the Mathieu group \(M_{12}\); the author also describes a construction of one of the graphs in terms of the Witt design \(S(5,8,24)\).