On the Existence of Triangular Difference Systems of Sets

A difference system of sets (DSS) is any collection of subsets of \(\mathbb{Z}_n\) with the property that the differences from distinct sets cover \(\mathbb{Z}_n\). That is, every non-zero class in \(\mathbb{Z}_n\) can be written as a difference of classes in at least one way. DSS were introduced by Levenstein in 1971 only for finite fields but the case for just 2 subsets had been previously considered by Clauge. Their work emphasized an application to synchronizable codes. A DSS is triangular if its sets contain only triangular numbers mod \(n\). We show that a triangular DSS cannot exist in \(\mathbb{Z}_{2^k}\) for \(k > 3\).