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In this paper, we exploit some intersection matrices to empower a
backtracking approach based on Kramer-Menser matrices. As an
application, we consider the interesting family of simple
$t-(t+8, t+2, 4)$ designs $1\leq t \leq 4$, and provide a complete
classification for $t=1, 4$ as well as a classification of all
non-rigid designs for $t=2, 3$. We also enumerate all rigid
designs for $t=2$. The computations confirm the results obtained
in Denny and Mathon [4] through the new approach which is much
simpler. Finally a list of other designs constructed by this
method is provided.
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