Note on parameters of balanced ternary designs

Conditions under which constant block size implies constant replication number and vice versa are established for balanced ternary designs. This note is motivated by a note of almost the identical title of W. D. Wallis [2]. In a balanced binary design with constant block size, the replication number must be constant, but the converse is not true. Wallis [2] proved the following partial converse: Theorem 1. If the positive integers v, b, r, and 'A are such that vr = bk and 'A(v-1) = r(k-1) for some integer k, and if there is an (r, 'A)design on v treatments with b blocks, then the design has constant block size k. We assume that the reader is familiar with the definition of ternary design and related notation including non-equireplicate, balanced ternary designs. See for example Billington [1]. A balanced ternary design with constant block size does not imply constant replication. For example, consider the balanced ternary design 112, 134, 134, 223, 224 with A = 2 and block size K = 3 but whose replication number is not constant. On the other hand, if we have a desig n with constant replication including Pi, the number of times an element occurs singly, and P2, the number of times an element occurs doubly, this does not imply constant block size. For example the balanced ternary design 1125, 1344, 13, 224, 2335, 455 is a balanced ternary design with A = 2 and the replication number R = 4 , Pi = 2, P2 = 1 but the block size is not constant. We prove the following result which is interesting in the sense that similar conditions like the binary case are required in the ternary case for constant Pi and P2 to imply constant block size, but 011e more condition is required to prove the converse. Australasian Journal of Combinatori cs 11 ( 1995) I pp. 177-179 Theorem 2. Suppose the positive integers V, B. R==P1 +2P2. K. A are such that VR == BK (1) A(V-1) == Pi (K-1) + 2P2(K-2) (2) and suppose there is a balanced ternary design on V elements, B blocks with balance A. If the balanced ternary design is equireplicate with constant P 1 and P 2 then it has constant block size K. Conversely, suppose in the balanced ternary design element i occurs ri1 times singly and ri2 times doubly and v I(ril +2ri2)2 == VR2; i=l if the design has constant block size then it is equireplicate with ri1 == P 1 and ri2 == P2 for each element i. Proof: Suppose that a balanced ternary design with replication number R and balance A exists, with Bj blocks of size Kj where i is from a finite index set. We have L:Bj==B L:BjKj== VR :LBiKj(Kj-1) == AV(V-1) + 2VP2 (3) (4) (5) From (1) and (4), the mean of the block sizes (L,BjKj)/(L:Bj) is K. From (4) and (5), L:BjK j2 == VR + AV(V-1) + 2VP2 (6) Now ~B'(K'-K)2 -~B·K·2 + ~B'K2 2~B'K'K 40.1 I -40.11 40.1 40.11 == L:BjK j2 -L:BjK2 == L:BiKj2 -BK2 == VR + AV(V-1) + 2VP2 VRK ( from (6) and (1) ) == V[P1(K-1) + 2P2(K-2)] VR(K-1) + 2VP2 == VR(K-1) 2VP2 VR(K-1) + 2VP2 == o. Therefore Kj == K for all i.