On existence of t-Designs with large ν and λ

Ray-Chaudhuri, D. K. ; Singhi, N. M. (1988) On existence of t-Designs with large ν and λ SIAM Journal on Discrete Mathematics, 1 (1). pp. 98-104. ISSN 0895-4801

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Related URL: http://dx.doi.org/10.1137/0401011

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 $\mu \leqq c_1 ( t,k )v ^{k - 2t}$.

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