On the Existence of Triangular Difference Systems of Sets

Abstract

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 $k>3$.