Lotto Designs

Let \(L(n, k, p, t)\) denote the minimum number of subsets of size \(k\) (\(k\)-subsets) of a set of size \(n\) (\(n\)-set) such that any \(p\)-subset intersects at least one of these \(k\)-subsets in at least \(t\) elements. The value of \(L(n, 6, 6, 2)\) is determined for \(n \leq 54\).