Semiframes with Block Size Three and Odd Group Size

A ( k , λ ) ‐semiframe of type g u is a ( k , λ ) ‐GDD of type g u , ( X , G , B ) , in which the collection of blocks B can be written as a disjoint union B = P ∪ Q where Q is partitioned into parallel classes of X and P is partitioned into holey parallel classes, each holey parallel class being a partition of X∖ G jfor someG j ∈ G . A ( k , λ ) ‐SF ( p , d , g u ) is a ( k , λ ) ‐semiframe of type g u in which there are p parallel classes in Q and d holey parallel classes with respect toG j ∈ G . In this paper, we shall show that there exists a (3, 1)‐SF ( p , d , g u ) for any g ≡ 3 ( mod 6 ) if and only if u ≥ 5 , u ≡ 1 ( mod 2 ) , p ≡ u − 1 2( mod u − 1 ) , d = g 2 − p u − 1 , and ( p , d , g , u ) ≠ ( 2 , 1 , 3 , 5 ) .