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Designs, Codes and Cryptography

, Volume 76, Issue 3, pp 551–569 | Cite as

Diagonally cyclic equitable rectangles

  • Anthony B. Evans
  • David Fear
  • Rebecca J. Stones
Article

Abstract

An equitable \((r,c;v)\)-rectangle is an \(r \times c\) matrix \(L=(l_{ij})\) with symbols from \(\mathbb {Z}_v\) in which each symbol appears in every row either \(\left\lceil c/v \right\rceil \) or \(\left\lfloor c/v \right\rfloor \) times and in every column either \(\left\lceil r/v \right\rceil \) or \(\left\lfloor r/v \right\rfloor \) times. We call \(L\) diagonally cyclic if \(l_{(i+1) (j+1)}=l_{ij}+1\), where the rows are indexed by \(\mathbb {Z}_r\) and columns indexed by \(\mathbb {Z}_c\). We give a constructive proof of necessary and sufficient conditions for the existence of a diagonally cyclic equitable \((r,c;v)\)-rectangle.

Keywords

Equitable rectangle Latin square Latin rectangle  Orthogonal array 

Mathematical Subclass Classification

05B15 

Notes

Acknowledgments

Stones supported by NSFC grant 61170301. Stones was also partially supported by AARMS.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Anthony B. Evans
    • 1
  • David Fear
    • 2
  • Rebecca J. Stones
    • 2
    • 3
    • 4
  1. 1.Department of Mathematics and StatisticsWright State UniversityDaytonUSA
  2. 2.School of Mathematical SciencesMonash UniversityClaytonAustralia
  3. 3.Clayton School of Information TechnologyMonash UniversityClaytonAustralia
  4. 4.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada

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