Existence of resolvable optimal strong partially balanced designs

We shall refer to a strong partially balanced design SPBD(v,b,k;λ,0)SPBD(v,b,k;λ,0) whose b   is the maximum number of blocks in all SPBD(v,b,k;λ,0)SPBD(v,b,k;λ,0), as an optimal strong partially balanced design, briefly OSPBD(v,k,λ)OSPBD(v,k,λ). Resolvable strong partially balanced design was first formulated by Wang, Safavi-Naini and 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,3,1)ROSPBD(v,3,1). We show that there exists an ROSPBD(v,3,1)ROSPBD(v,3,1) for any v⩾3v⩾3 except v=6,12v=6,12.