Sesqui-arrays, a generalisation of triple arrays
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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 row and a column are (possibly different) non-zero constants. Deleting the condition on the letters commonto 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 ofcolumns is deleted. Thus all triple arrays are sesqui-arrays.In this paper we give three constructions for sesqui-arrays. The first gives (n+1) x n2 arrays on n(n+1) letters for n>1. (Such an 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 7 x 36 sesqui-array for 42 letters. We also give a construction for K x (K-1)(K-2)/2 sesqui-arrays onK(K-1)/2 letters. This construction uses biplanes. It starts with a block of a biplane and produces an array which satisfies 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.
Bailey , R A , Cameron , P J & Nilson , T 2018 , ' Sesqui-arrays, a generalisation of triple arrays ' , Australasian Journal of Combinatorics , vol. 71 , no. 3 , pp. 427-451 .
Australasian Journal of Combinatorics
© 2018, the Author(s), Australasian Journal of Combinatorics. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://ajc.maths.uq.edu.au/?page=get_volumes&volume=71
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