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• 1.
University of St Andrews, Scotland.
University of St Andrews, Scotland. Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
Sesqui-arrays, a generalisation of triple arrays2018In: The Australasian Journal of Combinatorics, ISSN 1034-4942, Vol. 71, no 3, p. 427-451Article in journal (Refereed)

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.

• 2.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Pseudo-Youden designs balanced for intersection2011In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 141, no 6, p. 2030-2034Article in journal (Refereed)

If the row-column intersections of a row-column design $\mathcal{A}$ form a balanced incomplete block design, then $\mathcal{A}$ is said to be \emph{balanced for intersection}. This property was originally defined for triple arrays by McSorley et al. (2005a), section 8, where an example was presented and questions of existence were raised and discussed. We give sufficient conditions for the class of balanced grids in order to be balanced for intersection,  and prove that a family of binary pseudo-Youden designs has this property.

• 3.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
University of St Andrews, Scotland.
Triple arrays from difference sets2017In: Journal of combinatorial designs (Print), ISSN 1063-8539, E-ISSN 1520-6610, Vol. 25, no 11, p. 494-506Article in journal (Refereed)

This paper addresses the question whether triple arrays can be constructed from Youden squares developed from difference sets. We prove that if the difference set is abelian, then having -1 as multiplier is both a necessary and sufficient condition for the construction to work. Using this, we are able to give a new infinite family of triple arrays. We also give an alternative and more direct version of the construction, leaving out the intermediate step via Youden squares. This is used when we analyse the case of non-abelian difference sets, for which we prove a sufficient condition for giving triple arrays. We do a computer search for such non-abelian difference sets, but have not found any examples satisfying the given condition.

• 4.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
Inner balance of symmetric designs2014In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 71, no 2, p. 247-260Article in journal (Refereed)

A triple array is a row-column design which carries two balanced incomplete block designs (BIBDs) as substructures. McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005), Section 8, gave one example of a triple array that also carries a third BIBD, formed by its row-column intersections. This triple array was said to be balanced for intersection, and they made a search for more such triple arrays among all potential parameter sets up to some limit. No more examples were found, but some candidates with suitable parameters were suggested. We define the notion of an inner design with respect to a block for a symmetric BIBD and present criteria for when this inner design can be balanced. As triple arrays in the canonical case correspond to SBIBDs, this in turn yields new existence criteria for triple arrays balanced for intersection. In particular, we prove that the residual design of the related SBIBD with respect to the defining block must be quasi-symmetric, and give necessary and sufficient conditions on the intersection numbers. This, together with our parameter bounds enable us to exclude the suggested triple array candidates in McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005) and many others in a wide search. Further we investigate the existence of SBIBDs whose inner designs are balanced with respect to every block. We show as a key result that such SBIBDs must possess the quasi-3 property, and we answer the existence question for all known classes of these designs.

• 5.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
Umeå University, Department of Mathematics and Mathematical Statistics.
Triple arrays and Youden squares2015In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 75, no 3, p. 429-451Article in journal (Refereed)

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 {Mathematical expression}. For triple arrays with {Mathematical expression} it is proven that they can never be constructed in this way, and for triple arrays with {Mathematical expression} 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.

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