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A detailed description of an improved version of backtracking
algorithms for finding $t$-designs proposed by G. B. Khosrovshahi
and the authors of this paper [J Combin Designs 10(2002), 180-194]
is presented. The algorithm is then used to determine all
5-(14,6,3) designs admitting an automorphism of order 13,11,or 7.
It is concluded that a 5-(14,6,3) design with an automorphism of
prime order $p$ exists if and only if $p=2,3,7,13$.
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