Triple arrays and Youden squares

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• 作者：Tomas Nilson ; Lars-Daniel ?hman
• 关键词：Triple array ; Double array ; Youden square ; Difference set ; SBIBD ; 05B05 ; 05B10 ; 05B30
• 刊名：Designs, Codes and Cryptography
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：75
• 期：3
• 页码：429-451
• 全文大小：728 KB
• 参考文献：1.Agrawal H.: Some methods of construction of designs for two-way elimination of heterogeneity. J. Am. Stat. Assoc. 61, 1153-171 (1966).
  2.Ionin Y.J., Shrikhande M.S.: Combinatorics of Symmetric Designs. Cambridge University Press, New York, ISBN 978-0-521-81833-9 (2006).
  3.McSorley J.P., Phillips N.C.K., Wallis W.D., Yucas J.L.: Double arrays, triple arrays and balanced grids. Des. Codes Cryptogr. 35, 21-5 (2005).
  4.McSorley J.P.: Double arrays, triple arrays and balanced grids with \(v=r+c-1\) . Des. Codes Cryptogr. 37, 313-18 (2005).
  5.Preece D.A., Wallis W.D., Yucas J.L.: Paley triple arrays. Australas. J. Comb. 33, 237-46 (2005).
  6.Raghavarao D., Nageswararao G.: A note on a method of construction of designs for two-way elimination of heterogeneity. Commun. Stat. 3, 197-99 (1974).
  7.Wallis W.D., Yucas J.L.: Note on Agrawal’s “designs for two-way elimination of heterogeneity- J. Comb. Math. Comb. Comput. 46, 155-60 (2003).
  
• 作者单位：Tomas Nilson (1)
  Lars-Daniel ?hman (2)
  
  1. Department of Science Education and Mathematics, Mid-Sweden University, Sundsvall, Sweden
  2. Department of Mathematics and Mathematical Statistics, Ume? University, Ume?, Sweden
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Coding and Information Theory
  Data Structures, Cryptology and Information Theory
  Data Encryption
  Discrete Mathematics in Computer Science
  Information, Communication and Circuits
  
• 出版者：Springer Netherlands
• ISSN：1573-7586

文摘

This paper addresses the question of when triple arrays can be constructed from Youden squares by removing a column together with the symbols therein, and then exchanging the role of columns and symbols. The scope of the investigation is limited to the standard case of triple arrays with \(v=r+c-1\). For triple arrays with \(\lambda _{cc}=1\) it is proven that they can never be constructed in this way, and for triple arrays with \(\lambda _{cc}=2\) it is proven that there always exists a suitable Youden square and a suitable column for this construction. Further, it is proven that Youden square constructed from a certain family of difference sets never give rise to triple arrays in this way but always gives rise to double arrays. Finally, it is proven that all triple arrays in the single known infinite family, the Paley triple arrays, can all be constructed in this way for some suitable choice of Youden square and column.