Mixed Steiner Triples Systems with Shortest Length

TL;DR

This paper establishes the existence of shortest-length mixed Steiner triple systems with specific parameters, expanding the known constructions for these combinatorial designs.

Contribution

It proves the existence of minimal-length 3-GDDs for certain parameter sets and provides additional constructions for these systems.

Findings

Existence of 3-GDDs for specified parameters

Construction methods for shortest-length systems

Parameter conditions for design existence

Abstract

We prove that a 3-GDD of type $1^n k^1 \ell^1$, where $n= k \cdot \ell$, with minimum distance 3 exists for every $k$ and $\ell$ such that $n = k \ell$, $k = 1$ or $3~(mod ~ 6)$, and $\ell = 1$ or $3~(mod ~ 6)$. These designs are of the shortest possible length (smallest number of elements) for given $k$ and $\ell$. Other constructions for such triple systems are also presented.