Nuclear Designs

A Nuclear Design \(ND(v; k, \lambda)\) is a collection \( {B}\) of \(k\)-subsets of a \(v\)-set \(V\), where \( {B} = \mathcal{P}\cap {C} \), where \((V, \mathcal{P})\) is a maximum packing \((PD(v; k,\lambda))\) and \((V, \mathcal{C})\) is a minimum covering \((CD(v; k,\lambda))\) with \(|{B}|\) as large as possible. We construct \(ND(v; 3, 1)\)’s for all \(v\) and \(\lambda\). Along the way we prove that for every leave (excess) possible for \(k = 3\), all \(v,\lambda\), there is a maximum packing (minimum covering) achieving this leave (excess).