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On the spectrum of r-orthogonal Latin squares of different orders

فصلنامه معادلات در ترکیبات، دوره: 5، شماره: 2
Year: 1395
COI: JR_COMB-5-2_005
Language: EnglishView: 34
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Hanieh Amjadi - Alzahra University
Nasrin Soltankhah - Alzahra University
Naji Shajarisales - Max Planck Institute for Intelligent Systems
Mehrdad Tahvilian - Sharif University of Technology


‎Two Latin squares of order n are orthogonal if in their superposition‎, ‎each of the n^{۲} ordered pairs of symbols occurs exactly once‎. ‎Colbourn‎, ‎Zhang and Zhu‎, ‎in a series of papers‎, ‎determined the integers r for which there exist a pair of Latin squares of order n having exactly r different ordered pairs in their superposition‎. ‎Dukes and Howell defined the same problem for Latin squares of different orders n and n+k‎. ‎They obtained a non-trivial lower bound for r and solved the problem for k \geq \frac{۲n}{۳} ‎. ‎Here for k < \frac{۲n}{۳}‎, ‎some constructions are shown to realize many values of r and for small cases (۳\leq n \leq ۶)‎, ‎the problem has been solved‎.


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Amjadi, Hanieh and Soltankhah, Nasrin and Shajarisales, Naji and Tahvilian, Mehrdad,1395,On the spectrum of r-orthogonal Latin squares of different orders,

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  • G. B. Belyavskaya, r-Orthogonal quasigroups I, Math. Issled., ۳۹ (۱۹۷۶) ...
  • G. B. Belyavskaya, r-Orthogonal quasigroups II, Math. Issled., ۴۳ (۱۹۷۶) ...
  • G. B. Belyavskaya, r-Orthogonal Latin squares, in: J. Dénes and ...
  • C. J. Colbourn and L. Zhu, The spectrum of r-orthogonal ...
  • P. Dukes and J. Howell, The orthogonality spectrum for Latin ...
  • J. Howell, The intersection problem and different pairs problem for ...
  • H. J. Ryser, A combinatorial theorem with an application to ...
  • L. Zhu and H. Zhang, A few more r-orthogonal Latin ...
  • L. Zhu and H. Zhang, Completing the spectrum of r-orthogonal ...

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