Further results on optimal (v,4,2,1)(v,4,2,1)-OOCs

Let Φ(v,k,λa,λc)Φ(v,k,λa,λc) denote the maximum possible size among all (v,k,λa,λc)(v,k,λa,λc)-OOCs. A (v,k,λa,λc)(v,k,λa,λc)-OOC is said to be optimal   if its size is equal to Φ(v,k,λa,λc)Φ(v,k,λa,λc). In this paper, the constructions and the sizes of optimal (v,4,2,1)(v,4,2,1)-OOCs are investigated. An upper bound for Φ(v,4,2,1)Φ(v,4,2,1) is improved. The exact value of Φ(v,4,2,1)Φ(v,4,2,1) with v≤201v≤201 is given with the aid of computer search. An optimal (24hv,4,2,1)(24hv,4,2,1)-OOC with h∈{1,2}h∈{1,2} and v=p1p2⋯prv=p1p2⋯pr, where prime pi≡1(mod6) is constructed recursively. The existence of gg-regular (gp,4,2,1)(gp,4,2,1)-OOCs for g=3,6,9,16g=3,6,9,16, and pp 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)(v,4,2,1)-OOCs are obtained. In particular, Φ(v,4,2,1)=U(v)Φ(v,4,2,1)=U(v) for positive integer v≡80,400(mod480).