Dimension of the space generated by Steiner systems. Issue 5 (21st January 2019) - Ghodrati, Amir Hossein
Abstract: To every Steiner system with parameters ( t, k, v ) on 1, …, v and blocks B, we can assign its characteristic vector χ B, which is a ( 0, 1 ) ‐vector whose entries are indexed by the k ‐subsets of 1, …, v such that for each k ‐subset A of 1, …, v , χ B ( A ) = 1 if and only if A ∈ B . In this paper, we show that the dimension of the vector space generated by all of the characteristic vectors of Steiner systems with parameters ( t, k, v ) is ( v k ) − ( v t ) + 1, provided that 1 ≤ t ≤ k ≤ v / 2 and there is at least one such system.