On Existence of t-Designs with Large $v $ and $\lambda $

Abstract

It is shown that for $v $ sufficiently large and $k\geqq 2t + 1$, for any feasible quadruple $t - ( v ,k,\lambda )$ there exists a $t - ( v ,k,\lambda )$-design in which multiplicity of every block is 0 or $ \pm 1$ and the number of blocks with nonzero multiplicity is not too large compared to $\lambda \begin{pmatrix} v \\ t \end{pmatrix} $. As a consequence it is shown that the usual $t - ( v ,k,\lambda )$-designs in which no block is repeated more than twice exist if \[ \begin{pmatrix} v - t \\ k - t \end{pmatrix} + c_1 ( t,k )v ^{k - 2t} \geqq \lambda \geqq \begin{pmatrix} v - t \\ k - t \end{pmatrix} - c_1 ( t,k )v ^{k - 2t} \] where $c_1 ( t,k )$ is some function of t and k only. This implies that in Wilson’s result on the existence of a $t - ( v ,k,\lambda )$-design for \[ \lambda = m \begin{pmatrix} {v - t} \\ {k - t} \end{pmatrix} + \mu ,\quad 0\leqq \mu < \begin{pmatrix} {v - t} \\ {k - t} \end{pmatrix}, \] and m sufficiently large, the condition sufficiently large m can be replaced by $m\geqq 0$ when $\mu \geqq \begin{pmatrix} {v - t} \\ {k - t} \end{pmatrix} - c_1 ( t,k )v^{k - 2t} $ and by $m\geqq 1$ when $\mu \leqq c_1 ( t,k )v ^{k - 2t} $.