Once more about 80 Steiner triple systems on 15 points

✍ Scribed by Michel Deza; Viatcheslav Grishukhin

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 174 KB
• Volume: 72
• Category: Article
• ISSN: 0378-3758
• DOI: 10.1016/s0378-3758(98)00030-5

✦ Synopsis

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 Z v . 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; ) design, being identiÿed 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 ÿnd that for the 5 types R(L) = ∅, A 7 1 ; A2A 3 3 ; A6A7 and A14. The family with R(L) = ∅ contains only one STS, which is the projective space PG(3; 2). The family with R(L) = A 7 1 contains also only one STS. Two-graphs related to both the STSs belong to a family of two graphs discovered by T.