Mixed Steiner Triple Systems With Shortest Length

A mixed Steiner triple system is a 3-GDD which is viewed as a code with minimum Hamming distance 3. These codes are the minimum weight codewords of a 1-perfect code over a mixed alphabet, when the related codes exist, and provide the connection between 3-GDDs and coding theory. 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 $n=k\cdot \ell$, $k\equiv 1$ or $3\left(\text{mod}6\right)$, $\ell \equiv 1$ or $3\left(\text{mod}6\right)$, and $k\ge \ell >1$. 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.