IJPAM: Volume 54, No. 4 (2009)

GROUP DIVISIBLE DESIGNS WITH
TWO ASSOCIATE CLASSES AND $\lambda_1-\lambda_2=1$

Wannee Lapchinda$^1$, Nittiya Pabhapote$^2$
$^{1,2}$School of Science
University of the Thai Chamber of Commerce
126/1 Vibhavadee-Rangsit Road, Dindaeng, Bangkok, 10400, THAILAND
$^1$e-mail: wannee_lap@utcc.ac.th
$^2$e-mail: nittiya_pab@utcc.ac.th


Abstract.A group divisible design GDD $(v, g, k, \lambda_1, \lambda_2)$ is a collection of $k$-subsets (called blocks) of a $v$-set of symbols where: the $v$-set is divided into $g$ groups; each pair of symbols from the same group occurs in exactly $\lambda_1$ blocks; and each pair symbols from different groups occurs in exactly $\lambda_2$ blocks. Pairs of symbols occurring in the same group are known to statisticians as first associates, and pairs occurring in different groups are called second associates. The existence of such GDDs has been of interest over the years, going back to at least the work of Bose and Shimamoto in 1952 who began classifying such designs. Recently, such an existence problem when $g=2$ was solved in the case where the groups have the same size and the blocks have size 3. In this paper, we continue to focus on blocks of size 3, solving the problem when the required designs having two groups of unequal sizes and $\lambda_1-\lambda_2=1$ and prove that the conditions are sufficient.

Received: July 10, 2009

AMS Subject Classification: 05B05, 05B07

Key Words and Phrases: group divisible designs

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 54
Issue: 4