No extendable biplane of order nine
A t-(v,k,\(\lambda)\) design is a v-set S provided with a collection of k- subsets of s, called blocks, such that any t-subset of S is contained in exactly \(\lambda\) blocks. Using coding theoretic arguments, the author proves the nonexistence of a 3-(57,12,2) design. Together with a result of \textit{P. J. Cameron} [J. Comb. Theory, Ser. A 14, 215-220 (1973; Zbl 0252.05010)] and the nonexistence of a 3-(112,12,1) design [\textit{C. W. H. Lam}, \textit{L. Thiel}, \textit{S. Swiercz}, and \textit{J. Mckay}, Discrete Math. 45, 319-321 (1983; Zbl 0574.05008)] this shows that the only extendable biplane is the one of order three.
scientific article; zbMATH DE number 4081570