RESEARCH REPORT

The fine structure of (v; rho_2; 3, 3) balanced ternary designs with small v and rho_2 = 1, 2.

Authors: Peter Adams, Darryn E. Bryant and A. Khodkar

Centre for Discrete Mathematics and Computing, Department of Mathematics, The University of Queensland, Queensland 4072, Australia

Research supported by Australian Research Council grant A49532750

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 any one block, each pair of non-distinct elements, {x, x}, occurs in rho_2 blocks of the design and each pair of distinct elements, {x, y}, occurs lambda times throughout the design. We denote these parameters by (v; rho_2; k, lambda) BTD.

In this report k = lambda = 3 and rho_2 = 1, 2. Thus all blocks are of the form {x, y, z} or {x, x, y}. Note that the block {x, x, y} contains the pair {x, y} twice and the pair {x, x} once.

In [1] we prove the following result:

Main Theorem

(i) Fine_1(v) = Adm_1(v) for all v equiv 3 (mod 6), v \not \in {9,15 },

Fine_1(9) = Adm_1(9) \setminus {(0,4),(1,1)},

Adm_1(15) \setminus {(0,4)} \subseteq Fine_1(15) and

Fine_1(15) \subseteq Adm_1(15).

(ii) Fine_2(v) = Adm_2(v) for all v\equiv 3 (mod 6), v >= 9.

In this report we complete this by dealing with some small cases. Note that $c_2$ and $c_3$ denote the number of doubly and triply repeated blocks, respectively.

[1] Peter Adams, Darryn E. Bryant and A. Khodkar, The fine structure of balanced ternary designs with block size three, index three and rho_2=1, 2, Ars Combinatoria, (to appear).