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 [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). We show that there exists an ROSPBD(v, 3, 1) for any v≥3 except v = 6, 12.

We prove that there exists a pair of orthogonal diagonal Latin squares of order v with missing subsquares of side n (ODLS (v, n ) for all v≥3n+2 and v-n even. Further, there exists a magic square of order v with missing subsquare of side n (MS (v, n)) for all v≥3n+2 and v-neven.

Splitting balanced incomplete block designs were first formulated by Ogata, Kurosawa, Stinson, and Saido recently in the investigation of authentication codes. This article investigates the existence of splitting balanced incomplete block designs, i.e., (v, 3k, λ)-splitting BIBDs; we give the spectrum of (v, 3 × 2, λ)-splitting BIBDs. As an application, we obtain an infinite class of 2-splitting A-codes.

A Kirkman packing design KPD ({w, s*}, v) is a resolvable packing with maximum possible number of parallel classes, each class containing one block of size s and all other blocks of size w. A (t, w)-threshold scheme is a way of distributing partial information (shadows) to w participants, so that any t of them can easily calculate a key, but no subset of fewer than t participants can determine the key. In this paper we improve the existence results on KPD ({3, s*}, v) for s = 4, 5. We also obtain some results on KPD ({4, s*}, v) for s = 5, 6. These results can be used to give some new (2, w)-threshold schemes.

In this paper it is shown that an idempotent TD (5, m) − TD (5, n) exists whenever the known necessary condition m ≥ 4n+1 is satis3ed, except when (m, n)=(6, 1) and possible when (m, n)=(10, 1). For m ＜ 60 and n ≤ 10, we also indicate where several idempotent TD(k, m)−TD(k, n)’s for k = 6, 7 can be found.