The Existence of 3 2 -DSS ( 4 , 6 ) -GDDs of Type g u

Abstract

A design is said to be super-simple if any two of its blocks have at most two common points. A design with index $t\lambda$ is said to be ${\lambda }^t$-decomposable, if its blocks can be partitioned into nonempty collections ${ℬ}_i,1\le i\le t$, such that each ${ℬ}_i$ with the point set forms a design with index $\lambda$. In this paper, it is proved that there exists a $3^2$-DSS $\left(4,6\right)$-GDD (decomposable super-simple group divisible design) of type $g^u$ if and only if $u\ge 4,g\left(u-2\right)\ge 12$ and $g^{2}u\left(u-1\right)\equiv 0\left(\mathrm{mod}4\right)$, and except possibly for $g^u\in \left\{2^8,6^4\right\}$.