Notes on the Support of $t$-Designs

Abstract

The support of a $t$-design is the set of all distinct blocks in the design. The notation $t-(v,k,\lambda |b^{\ast })$ is used to denote a $t$-design with precisely $b^{\ast }$ distinct blocks. We present some results about the structure of support in $t$-designs. Some of them are about the number and the range of occurrences of $i$-sets ($1\le i\le t$) in the support. A new bound for the support sizes of $t$-designs is presented. In particular, given a $t-(v,k,\lambda |b^{\ast })$ design with $b>b_0$, where $b$ and $b_0$ are the cardinality and the minimum cardinality of block sets in the design, respectively, then it is shown that $b^{\ast }\ge \lceil \frac{\lceil \frac{2b}{\lambda }\rceil +7}{2}\rceil$. We also show that when $\lambda$ varies over all positive integers, then there is no $t-(v,k,\lambda |b^{\ast })$-design with the support sizes equal to $b_{min}^{\ast }+1,b_{min}^{\ast }+2$ and $b_{min}^{\ast }+3$, where $b_{min}^{\ast }$ denotes the least possible cardinality of the support sizes in this design.