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

## TL;DR

This paper establishes sufficient conditions based on the structure and orthogonal Latin squares for the resolvability of certain Steiner triple systems with specific 3-rank constraints, enabling their partition into parallel classes.

## Contribution

It provides new sufficient conditions for the resolvability of $STS(3^k)$ with limited 3-rank, linking their structure to orthogonal Latin squares and partitioning into parallel classes.

## Key findings

- Conditions for resolvability of $STS(3^k)$ are established.
- Block sets can be partitioned into parallel classes under these conditions.
- Resolvability is proved when conditions are satisfied.

## 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.