Cyclic near difference sets of type 1 (Q1313977)

Necessary conditions for the existence of \(k\)-sets \(D \subseteq \mathbb{Z}_ v\) such that the equation \(x-y=a\) \((x,y \in D)\) has exactly \(\lambda\) solutions for \(a\) different from 0, \(v/2\), no solution otherwise, are given. The number (often =1) of such sets \(D\) for given \((v,k,\lambda)\) are determined, mainly for the cases \(\lambda=1\); \(k=v/2\); \(k=(v/2)-1\). Complex constructions are given, manily using the union of orbits of multipliers of requested \(D\)'s.