Cyclic Difference Sets of Positive Deficiency
Martin Funk · CINECA IRIS Institutional Research Information System (University of Basilicata) · 2008
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 $i \ e j$, 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 $n_k$.