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Paper IPM / M / 149  


Abstract:  
We call a 2design D with parameters v,k, and
λ = ((v−2)  (k−2)) a complete design. The number of
distinct blocks of D, called the support size of
D, is denoted by b^{*}. For a complete design with v ≥ 7 and
for k=3, Constantine and Hedayat (J. Statist. Plann. Inference
7 (1993), 289294) have shown that max b^{*}=((v)  3)−4(v−3), provided a block of D attains the maximum
multiplicity, λ. In this paper, we show that if a block of
a complete design D with k ≥ 3 is repeated maximum
possible times (i.e.,λ = ((v−2)  (k−2))), then b^{*} ≤ ((v)  (k))−k [ ((v−1)  (k−1))−((v−2)  (k−2))−((v−k)  (k−1))]− ((v−2)  (k−2))+1. Furthermore,
for v=k^{2}−k+1, where k−1 is a prime power, and also for
v ≡ 1 (mod 12), where k=4, we construct designs for which
the equality for b^{*} holds.
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