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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Official URL: http://epubs.siam.org/sidma/resource/1/sjdmec/v1/i...

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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Deposited On: | 23 Jul 2011 12:05 |

Last Modified: | 23 Jul 2011 12:05 |

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