Sufficient conditions for STS$(3^k)$ of 3-rank $\leq 3^k-r$ to be resolvable

Abstract

Based on the structure of non-full-$3$-rank $STS(3^k)$ and the orthogonal Latin squares, we mainly give sufficient conditions for $STS(3^k)$ of $3$-rank $\leq 3^k-r$ to be resolvable in the present paper. Under the conditions, the block set of $STS(3^k)$ can be partitioned into $\frac{3^k-1}{2}$ parallel classes, i.e., $\frac{3^k-1}{2}$ $1$-$(v,3,1)$ designs. Finally, we prove that $STS(3^k)$ of 3-rank $\leq 3^k-r$ is resolvable under the sufficient conditions.