Combinatorial constructions for optimal two-dimensional optical orthogonal codes with λ = 2

In this paper, we are concerned about optimal two-dimensional optical orthogonal codes with $\lambda$ = 2. Some combinatorial constructions are presented and many infinite families of optimal two-dimensional optical orthogonal codes with weight 4 and $\lambda$ = 2 are obtained. Especially, we shall see that in many cases an optimal two-dimensional optical orthogonal code can not achieve the Johnson bound.