Menu
A balanced incomplete block design \(B[k, \alpha; v]\) is said to be a nested design if one can add a point to each block in the design and so obtain a block design \(B[k + 1, \beta; v]\). Stinson (1985) and Colbourn and Colbourn (1983) proved that the necessary condition for the existence of a nested \(B[3, \alpha; v]\) is also sufficient. In this paper, we investigate the case \(k = 4\) and show that the necessary condition for the existence of a nested \(B[4, \alpha; v]\), namely \(\alpha = 3\lambda\), \(\lambda(v – 1) \equiv 0 \pmod{4}\) and \(v \geq 5\), is also sufficient. To do this, we need the concept of a doubly nested design. A \(B[k, \alpha; v]\) is said to be doubly nested if the above \(B[k + 1, \beta; v]\) is also a nested design. When \(k = 3\), such a design is called a doubly nested triple system. We prove that the necessary condition for the existence of a doubly nested triple system \(B[3, \alpha; v]\), namely \(\alpha = 3\lambda\), \(\lambda(v – 1) \equiv 0 \pmod{2}\) and \(v \geq 5\), is also sufficient with the four possible exceptions \(v = 39\) and \(\alpha = 3, 9, 15, 21\).