A Principal Difference System and Arithmetic Progressions

A difference system of sets (DSS) is a collection of subsets of \(\mathbb{Z}_n\), the integers mod \(n\), with the property that each non-zero element of \(\mathbb{Z}_n\) appears at least once as the difference of elements from different sets. If there is just one set, it is called a principal DSS. DSS arise naturally in the study of systematic synchronizable codes and are studied mostly over finite fields when \(n\) is a prime power. Using only triangular numbers mod \(n\), we constructed a DSS over \(\mathbb{Z}_n\) for each positive integer \(n > 3\). Necessary and sufficient conditions are given for the existence of a principal DSS using only triangular numbers in terms of coverings of \(\{1, \ldots, n-1\}\) by finite arithmetic progressions.