There are exactly five biplanes with k=11

A biplane is a 2-(k(k − 1)/2 + 1, k, 2) symmetric design. Only sixteen nontrivial biplanes are known: there are exactly nine biplanes with k < 11, at least five biplanes with k = 11, and at least two biplanes with k = 13. It is here shown by exhaustive computer search that the list of five known biplanes with k = 11 is complete. This result further implies that there exists no 3-(57, 12, 2) design, no 11211 symmetric configuration, and no (324, 57, 0, 12) strongly regular graph. The five biplanes have 16 residual designs, which by the Hall–Connor theorem constitute a complete classification of the 2-(45, 9, 2) designs.