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Sesqui-arrays, including triple arrays
University of St Andrews.
University of St Andrews.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics. (Mathematics)
2017 (English)In: arXiv.orgArticle in journal (Other academic) Submitted
Abstract [en]

A triple array is a rectangular array containing letters, each letter occurring equally often with no repeats in rows or columns, such that the number of letters common to two rows, two columns, or a rowand a column are (possibly different) non-zero constants. Deleting the condition on the letters common to a row and a column gives a double array. We propose the term sesqui-array for such an array when only the condition on pairs of columns is deleted. In this paper we give three constructions for sesqui-arrays. Therst gives $(n + 1)\times n^2$ arrays on n(n + 1) letters for $n\geq 2$. (Suchan array for n = 2 was found by Bagchi.) This construction uses Latin squares. The second uses the Sylvester graph, a subgraph of the Hoffman--Singleton graph, to build a good block design for 36 treatments in 42 blocks of size 6, and then uses this in a 736 sesqui-array for 42 letters.We also give a construction for K(K-1)(K-2)/2 sesqui-arrays on K(K-1)/2 letters from biplanes. The construction starts with a block of a biplane and produces an array which satises the requirements for a sesqui-array except possibly that of having no repeated letters in a row or column. We show that this condition holds if and only if the Hussain chains for the selected block contain no 4-cycles. A sufficient condition for the construction to give a triple array is that each Hussain chain is a union of 3-cycles; but this condition is not necessary, and we give a few further examples. We also discuss the question of which of these arrays provide good designs for experiments.

Place, publisher, year, edition, pages
2017.
Keyword [en]
Sesqui array, triple array, biplane
National Category
Discrete Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-30900OAI: oai:DiVA.org:miun-30900DiVA: diva2:1111699
Projects
Construction methods for triple arrays
Available from: 2017-06-19 Created: 2017-06-19 Last updated: 2017-06-26

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Other links

https://arxiv.org/abs/1706.02930

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CiteExportLink to record
Permanent link

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Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf