Sesqui-arrays, a generalisation of triple arrays

Abstract

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 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. Thus all triple arrays are sesqui-arrays.In this paper we give three constructions for sesqui-arrays. The first gives (n + 1) × n 2 arrays on n(n + 1) letters for n ≥ 2. (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 × 36 sesquiarray for 42 letters.We also give a construction for K × (K − 1)(K − 2)/2 sesquiarrays on K(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 4cycles. 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.