Doubly Nested Triple Systems And Nested $B[4,3\lambda ;v]$

Abstract

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 $\alpha =3\lambda$, $\lambda (v–1)\equiv 0(\mathrm{mod}4)$ and $v\ge 5$, 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 $\alpha =3\lambda$, $\lambda (v–1)\equiv 0(\mathrm{mod}2)$ and $v\ge 5$, is also sufficient with the four possible exceptions $v=39$ and $\alpha =3,9,15,21$.