Mixed Steiner Triples Systems with Shortest Length

We prove that a 3-GDD of type $1^n{k}^1{\mathrm{\ell }}^1$, where $n=k\cdot \mathrm{\ell }$, with minimum distance 3 exists for every $k$ and $\mathrm{\ell }$ such that $n=k\mathrm{\ell }$, $k=1$ or $3(mod6)$, and $\mathrm{\ell }=1$ or $3(mod6)$. These designs are of the shortest possible length (smallest number of elements) for given $k$ and $\mathrm{\ell }$. Other constructions for such triple systems are also presented.