There are exactly five biplanes with k = 11

✍ Scribed by Petteri Kaski; Patric R. J. Östergård

• Book ID: 102308211
• Publisher: John Wiley and Sons
• Year: 2008
• Tongue: English
• Weight: 167 KB
• Volume: 16
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20145

✦ Synopsis

Abstract

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 112~11~ 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. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 117–127, 2008