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Designs, Codes and Cryptography

, Volume 87, Issue 4, pp 769–784 | Cite as

Fano Kaleidoscopes and their generalizations

  • Marco Buratti
  • Francesca MerolaEmail author
Article
  • 65 Downloads
Part of the following topical collections:
  1. Special Issue: Finite Geometries

Abstract

In this work we introduce Fano Kaleidoscopes, Hesse Kaleidoscopes and their generalizations. These are a particular kind of colored designs for which we will discuss general theory, present some constructions and prove existence results. In particular, using difference methods we show the existence of both a Fano and a Hesse Kaleidoscope on v points when v is a prime or prime power congruent to 1\(\pmod {6}\), \(v\ne 13\). In the Fano case this, together with known results on pairwise balanced designs, allows us to prove the existence of Kaleidoscopes of order v for many other values of v; we discuss what the situation is, on the other hand, in the Hesse and general case.

Keywords

Colored designs Difference families Cyclotomy Pairwise balanced designs 

Mathematics Subject Classification

05B05 05B10 05C15 

Notes

Acknowledgements

Many thanks are due to P. Östergård for his help in trying to find a Fano Kaleidoscope of order 13 (see Remark 5.3). We also thank the reviewers for their helpful comments.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di PerugiaPerugiaItaly
  2. 2.Dipartimento di Matematica e FisicaUniversità Roma TreRomeItaly

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