Geodesic
On Block-Schematic Steiner Systems: S(t, t + 1, v)
Canadian journal of mathematics, Tome 33 (1981) no. 6, pp. 1432-1438

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A Steiner system S(t, k, v) is a collection of k-subsets, called blocks, of a v-set of points with the property that any t-subset of points is contained in a unique block. We assume 1 < t < k < v. A Steiner system is called block-schematic if the blocks form an association scheme with the relations determined by size of intersection. Ito and Patton [3] proved that if S(4, 5, v) is block-schematic, then v = 11. The purpose of this paper is to extend this result, and we prove the following theorem.THEOREM. A Steiner system S(t, t + 1, v) is block-schematic if and only if one of the following holds: (i) t = 2, (ii) t = 3, v = 8, (iii) t = 4, v = 11, (iv) t = 5, v = 12. 
Yoshizawa, Mitsuo. On Block-Schematic Steiner Systems: S(t, t + 1, v). Canadian journal of mathematics, Tome 33 (1981) no. 6, pp. 1432-1438. doi: 10.4153/CJM-1981-109-x
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