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

Abstract

AbstractThe 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 $$N > 1$$ N > 1 there is a large set $$LS[N](t, t+1, t+N\cdot \ell (t))$$ L S [ N ] ( t , t + 1 , t + N · ℓ ( t ) ) , where $$\ell (t)=\prod _{i=1}^t \lambda (i)\cdot \lambda ^*(i)$$ ℓ ( t ) = ∏ i = 1 t λ ( i ) · λ ∗ ( i ) , $$\lambda (t)=\mathop {\textrm{lcm}}(\left( {\begin{array}{c}t\\ m\end{array}}\right) \,\vert \, m=1,2,\ldots , t)$$ λ ( t ) = lcm ( t m | m = 1 , 2 , … , t ) and $$\lambda ^*(t)=\mathop {\textrm{lcm}}(1,2, \ldots , t+1)$$ λ ∗ ( t ) = lcm ( 1 , 2 , … , t + 1 ) , we obtain the following statement. If $$(t+2)$$ ( t + 2 ) is composite, then there is a large set $$LS[N](t, t+2, t+1+N\cdot \ell (t))$$ L S [ N ] ( t , t + 2 , t + 1 + N · ℓ ( t ) ) for all $$N > 1$$ N > 1 . If $$(t+2)$$ ( t + 2 ) is prime, then there is an $$LS[N](t, t+2, t+1+N\cdot \ell (t))$$ L S [ N ] ( t , t + 2 , t + 1 + N · ℓ ( t ) ) for any N with $$\gcd (t+2,N)=1$$ gcd ( t + 2 , N ) = 1 .