Constructions for t-designs and s-resolvable t-designs

Abstract

The purpose of the present paper is to introduce recursive methods for constructing simple t-designs, s-resolvable t-designs, and large sets of t-designs. The results turn out to be very effective for finding these objects. In particular, they reveal a fundamental property of the considered designs. Consequently, many new infinite series of simple t-designs, t-designs with s-resolutions and large sets of t-designs can be derived from the new constructions. For example, by starting with an important result of Teirlinck stating that for every natural number t and for all \(N > 1\) there is a large set \(LS[N](t, t+1, t+N\cdot \ell (t))\), where \(\ell (t)=\prod _{i=1}^t \lambda (i)\cdot \lambda ^*(i)\), \(\lambda (t)=\mathop {\textrm{lcm}}(\left( {\begin{array}{c}t\\ m\end{array}}\right) \,\vert \, m=1,2,\ldots , t)\) and \(\lambda ^*(t)=\mathop {\textrm{lcm}}(1,2, \ldots , t+1)\), we obtain the following statement. If \((t+2)\) is composite, then there is a large set \(LS[N](t, t+2, t+1+N\cdot \ell (t))\) for all \(N > 1\). If \((t+2)\) is prime, then there is an \(LS[N](t, t+2, t+1+N\cdot \ell (t))\) for any N with \(\gcd (t+2,N)=1\).