Resolvable optimal strong partially balanced designs with block size four

We shall refer to a strong partially balanced design SPBD(v,b,k;λ,0) whose b is the maximum number of blocks in all SPBD(v,b,k;λ,0), as an optimal strong partially balanced design, briefly OSPBD(v,k,λ). Resolvable strong partially balanced design was first formulated by Wang, Safavi-Naini and Pei in 2001 [Y. Wang, R. Safavi-Naini, D. Pei, Combinatorial characterization of l-optimal authentication codes with arbitration, J. Combin. Math. Combin. Comput. 37 (2001) 205-224] in investigation of l-optimal authentication codes. This article investigates the existence of resolvable optimal strong partially balanced design ROSPBD(v,4,1). We show that there exists an ROSPBD(v,4,1) for any v≡0,1,3,4,5,6,8, or 9(mod12) except v=8,12,20 and some possible exceptions.