Mixed Steiner Triple Systems With Shortest Length

ABSTRACT

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 , where , with minimum distance 3 exists for every and such that , or , or , and . These designs are of the shortest possible length (smallest number of elements) for given and . Other constructions for such triple systems are also presented.