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.

Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Kp,q factorizeation of Km,n is a set of edge-disjoint Kp,q-factors of Km,n which partition the set of edges of Km,n. When p=1 and q is a prime number, Wang, in his paper “On K1,k-factorizations of a complete bipartite graph” (Discrete Math, 1994, 126: 359-364), investigated the K1,q-factorization of Km,n and gave a sufficient condition for such a factorizeation to exist. In the paper “K1,k-factorizations of complete bipartite graphs” (Discrete Math, 2002, 259: 301-306), Du and Wang extended Wang’s result to the case that q is any positive integer. In this paper, we give a sufficient condition for Km,n to have a Kp,q-factorization. As a special case, it is shown that the Martin’s BAC conjecture is true when p:q=k:(k+1) for any positive integer k.

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.

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 shall refer to a diagonal Latin square which is orthogonal to its (3, 2, 1)-conjugate, and the latter is also a diagonal Latin square, as a (3, 2, 1)-conjugate orthogonal diagonal Latin square, briefly CODLS. This article investigates the spectrum of CODLS with a missing sub-square. The main purpose of this paper is two-fold. First of all, we show that for any positive integers n ≥ 1, a CODLS of order v with a missing subsquare of order n exists if v ≥ 13n/4 + 93 and v − n is even. Secondly, we show that for 2 ≤ n ≤ 6, a CODLS of order v with a missing subsquare of order n exists if and only if v ≥ 3n+2 and v − n is even, with one possible exception.