On minimal blocking sets in non-symmetric designs

Abstract

Let S be a blocking set of a 2−(v, k, λ) design D. In [13] and in [8] it is proved that (v−Smax ) ≤ |S| ≤ Smax = [vk + (k 2 v 2 − 4kv 2 + 4kv)1/2]/(2k) and k ≥ 4. Moreover, |S| ∈ {v − Smax , Smax ) if and only if S is of type (1, k − 1). In this paper, we deal with minimal blocking sets in a 2−(v, k, λ) design. We prove that a blocking set of type (1, k −1) is necessarily minimal. Hence, in a design (that we call max-blocked) containing a blocking set of type (1, k, − 1) the number Smax is also the maximum possible cardinality for a minimal blocking set. We prove that if S is a minimal blocking set of a 2−(v, k, λ) design D, such that |B ⋂ S| ≤ m for each block B, then |S| ≤ mb/(r + m −1). Hence it holds that |S| ≤ b(k − 1)/(r + k − 2). Moreover, we prove that there are only two families of designs in which the previous equality holds.