A code is said to be equidistant if the distance between any two distinct codewords of the code is the same. In this paper, we have studied equidistant single-orbit cyclic and quasi-cyclic subspace codes. The orbit code generated by a subspace U in \({\mathbb {F}}_{q^n}\) such that the dimension of U over \({\mathbb {F}}_q\) is t or \(n-t\) , \(\text{ where }~t=\dim _{{\mathbb {F}}_q}(\text{ Stab }(U)\cup \{0\})\) , is equidistant and is termed a trivial equidistant orbit code. Using the concept of cyclic difference sets, we have proved that only the trivial equidistant single-orbit cyclic subspace codes exist. Further, we have explored equidistant single-orbit quasi-cyclic subspace codes, focusing specifically on those which are sunflowers.
