Cyclic Difference Sets of Positive Deficiency

A subset $S = \{s_1, \ldots, s_k\} \subseteq \mathbb{Z}_n$ is called a {\it cyclic difference set modulo} $n$ {\it of order} $k$ {\it and deficiency} $\delta = n - k^2 + k - 1$ if, for $i,j = 1, \ldots, k$ with $i \ne j$, the $k^2 - k$ differences $ s_i - s_j \; (\makebox{mod} \, n) $ are pairwise distinct. Planar cyclic difference sets provide instances having deficiency $0$, whereas $k$--mark Golomb rulers produce infinitely many examples of cyclic difference sets modulo $n$ of order $k$ and positive deficiency, for all $n \ge 2L_k+1$ where $L_k$ denotes the length of an optimal $k$--mark Golomb ruler. We present two constructions which yield deficient difference sets modulo $n$ with $n \le 2L_k$. As an application, these results fill some gaps in the spectrum of cyclic configurations of type $n_k$.