Various super-simple designs with block size four

In this note the existence of a (Vj P2j 4, 2) BTD I for P2 = 0, 1 and 2, in which any pair of blocks intersect in at most two elements, is proved for any admissible v. A balanced ternary design is a collection of multi-sets of size k, chosen from a v-set in such a way that each element occurs 0, 1 or 2 times in anyone block, each pair of non-distinct elements, {x, x}, occurs in P2 blocks of the design and each pair of distinct elements, {x, y}, occurs , X times throughout the design. We denote these parameters by (v; P2i k,'x) BTD. It is easy to see that each element must occur singly in a constant number of blocks, say PI blocks, and so each element occurs altogether r = PI + 2P2 times. Also if b is the number of blocks in the design, then vr bk and ,X(v-1) = r(k-1)-2p2. (For further information [3] should be consulted.) A BTD is called simple if it contains no repeated blocks. A (Vi P2j 4, ,X) BTD is said to be super-simple if any pair of its blocks have at most two elements in common, where repetition of elements is counted. For example, the blocks xxyz and xxst are said to have two elements in common. Obviously, any super-simple BTD is a simple BTD. In [7] Gronau and Mullin introduce super-simple (Vj OJ 4,),) BTDs (which are of course balanced incomplete block designs (v, 4, ,X)) and in [9] Kejun proves that super-simple (Vi OJ 4, 3) BTDs exist if and oniy if v == 0 or 1 In this note we concentrate on the cases P2 = 0, 1 and 2, k = 4 and , X = 2. Indeed, we shall prove the following results. MAIN THEOREM (1) There exists a super-simple (v, 4,2) BIBD if and only if v == 1 (mod 3) and v-=F 4 ([7], Theorem A). (2) There exists a super-simple (Vj 1; 4, 2) BTD if and only if v == 0 (mod 6). (3) There exists a super-simple (v; 2; 4, 2) BTD if and only if v == 2 (mod 3), v ;;:: 11.