A class of symmetric association schemes as inclusion of biplanes

Let ${\cal B}$ be a nontrivial biplane of order $k-2$ represented by symmetric canonical incidence matrix with trace $1+ \binom{k}{2}$. We proved that ${\cal B}$ includes a partially balanced incomplete design with association scheme of three classes. Consequently, these structures are symmetric, having $2k-6$ points. While it is not known whether this class is finite or infinite, we show that there is a related superclass with infinitely many representatives.Comment: 11 pages, 2 figure

Non-transversal Vectors of Some Finite Geometries

By means of associated structural invariants, we efficiently construct four biplanes of order 9 - except the one with the smallest automorphism group, that is found by Janko and Trung. The notion of non-transversal vector is introduced since we observed related properties that provide significantly more efficient constructions. There is a dichotomy in the structure of biplanes of order 7 and 9 with respect to the incidence matrix symmetry.Comment: 12 page