Once more about 80 Steiner triple systems on 15 points

Abstract Subsets of a v-set are in one-to-one correspondence with vertices of a v-dimensional unit cube, a Delaunay polytope of the lattice Zv. All vertices of the same cardinality k generate a (v−1)-dimensional root lattice Av−1 and are vertices of the Delaunay polytope P(v,k) of the lattice Av−1. Hence k-blocks of a t−(v,k,λ) design, being identified with vertices of P(v,k), generate a sub-lattice of Av−1. We show that 80 Steiner triple systems (STS for short) 2-(15,3,1) are partitioned into 5 families. STSs of the same family generate the same lattice L. Each lattice L is distinguished by a set R(L) of its vectors of norm 2. R(L) is a root system. We find that for the 5 types R(L)=∅, A 1 7 , A 2 A 3 3 , A 6 A 7 and A14. The family with R(L)=∅ contains only one STS, which is the projective space PG(3,2). The family with R(L)=A17 contains also only one STS. Two-graphs related to both the STSs belong to a family of two graphs discovered by T. Spence and described by Seidel (1992, More about two-graphs, Nesetril, J. and Fiedler, M. (Eds.), Elsevier, Amsterdam, pp. 287–308).