Further results on optimal -OOCs

Let Φ ( v , k , λ a , λ c ) denote the maximum possible size among all ( v , k , λ a , λ c ) -OOCs. A ( v , k , λ a , λ c ) -OOC is said to be optimal if its size is equal to Φ ( v , k , λ a , λ c ) . In this paper, the constructions and the sizes of optimal ( v , 4 , 2 , 1 ) -OOCs are investigated. An upper bound for Φ ( v , 4 , 2 , 1 ) is improved. The exact value of Φ ( v , 4 , 2 , 1 ) with v ≤ 201 is given with the aid of computer search. An optimal ( 24 h v , 4 , 2 , 1 ) -OOC with h ∈ { 1 , 2 } and v = p 1 p 2 ⋯ p r , where prime p i ≡ 1 ( mod 6 ) is constructed recursively. The existence of g -regular ( g p , 4 , 2 , 1 ) -OOCs for g = 3 , 6 , 9 , 16 , and p a prime satisfying a suitable congruence is established by direct constructions. Furthermore, the sizes of several new infinite classes of optimal ( v , 4 , 2 , 1 ) -OOCs are obtained. In particular, Φ ( v , 4 , 2 , 1 ) = U ( v ) for positive integer v ≡ 80 , 400 ( mod 480 ) .