RETRACTED ARTICLE: Constructions for t-designs and s-resolvable t-designs

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 L S [ N ] ( t , t + 1 , t + N · ℓ ( t ) ) , where ℓ ( t ) = ∏ i = 1 t λ ( i ) · λ ∗ ( i ) , λ ( t ) = lcm ( t m | m = 1 , 2 , … , t ) and λ ∗ ( t ) = lcm ( 1 , 2 , … , t + 1 ) , we obtain the following statement. If ( t + 2 ) is composite, then there is a large set L S [ N ] ( t , t + 2 , t + 1 + N · ℓ ( t ) ) for all N > 1 . If ( t + 2 ) is prime, then there is an L S [ N ] ( t , t + 2 , t + 1 + N · ℓ ( t ) ) for any N with gcd ( t + 2 , N ) = 1 .