The four known biplanes with k = 11

ABSTRACT. The four known biplanes of order 9 (k ii) are described in terms of their ovals, %-chain structures, and automorphism groups. An exhaustive computer search for all biplanes of order 9 with certain chain structures has produced but two, one of which is new. None of these four biplanes yield the putative plane of order i0. Only finitely many biplanes (projective designs with % 2) are presently known and it is conjectured that for any fixed %> 2 but finitely many projective designs exist. The authors know of four biplanes of order 9. All are self-dual. The most symmetric of these has repeatedly been uncovered. A group-theoretica