Partitions of the 4-subsets of a 13-set into disjoint projective planes

Abstract For 1⩽ t k v , let S ( t , k , v ) denote a Steiner system and let P k ( v ) be the set of all k -subsets of the set {1,2,…, v }. We partition P 4 (13) into 55 mutually disjoint S (2, 4, 13)'s (projective planes). This is the first known example of a complete partition of P k ( v ) into disjoint S ( t , k , v )'s for k ⩾4 and t ⩾2.