Given positive integers v, k, t and \(\lambda \) with \(v \ge k \ge t\) , a packing design \(\textrm{PD}_{\lambda }(v,k,t)\) is a pair \((V,\mathcal{B})\) , where V is a v-set and \(\mathcal{B}\) is a collection of k-subsets of V such that each t-subset of V appears in at most \(\lambda \) elements of \(\mathcal{B}\) . When \(\lambda =1\) , a \(\textrm{PD}_1(v,k,t)\) is equivalent to a binary code with length v, minimum distance \(2(k-t+1)\) and constant weight k. The maximum size of a \(\textrm{PD}_{\lambda }(v,k,t)\) is called the packing number, denoted \(\textrm{PDN}_{\lambda }(v,k,t)\) . In this paper we consider packing designs with k large relative to v. We prove that for a positive integer n, \(\textrm{PDN}_{\lambda }(v,k,t) = n\) whenever \(nk-(t-1)\left( {\begin{array}{c}n\\ \lambda +1\end{array}}\right) \le \lambda v < (n+1)k-(t-1)\left( {\begin{array}{c}n+1\\ \lambda +1\end{array}}\right) \) . We also prove that if no point appears in more than three blocks, then the blocks of a \(\textrm{PD}_2(v,k,2)\) can be ordered so that no ordered pair occurs more than once. This produces a directed packing design and we show that the corresponding directed packing number is equal to n when \(nk-\left( {\begin{array}{c}n\\ 3\end{array}}\right) \le 2v < (n+1)k-\left( {\begin{array}{c}n+1\\ 3\end{array}}\right) \) . Such directed packing designs yield \((k-t)\) -deletion-correcting codes.
