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 $n_k$.

Cyclic Difference Sets of Positive Deficiency

FUNK, Martin
2008-01-01

Abstract

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 $n_k$.
2008
File in questo prodotto:
File Dimensione Formato  
CDS-ICA-rev-v1.pdf

non disponibili

Tipologia: Documento in Pre-print
Licenza: DRM non definito
Dimensione 192.46 kB
Formato Adobe PDF
192.46 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11563/435
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact