A Principal Difference System and Arithmetic Progressions

Abstract

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 $\{1,\dots ,n-1\}$ by finite arithmetic progressions.