Uniform semi-Latin squares and their pairwise-variance aberrations
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For integers n > 2 and k > 0, an (n×n)∕k semi-Latin square is an n × n array of k-subsets (called blocks) of an nk-set (of treatments), such that each treatment occurs once in each row and once in each column of the array. A semi-Latin square is uniform if every pair of blocks, not in the same row or column, intersect in the same positive number of treatments. It is known that a uniform (n × n)∕k semi-Latin square is Schur optimal in the class of all (n × n)∕k semi-Latin squares, and here we show that when a uniform (n × n)∕k semi-Latin square exists, the Schur optimal (n × n)∕k semi-Latin squares are precisely the uniform ones. We then compare uniform semi-Latin squares using the criterion of pairwise-variance (PV) aberration, introduced by J. P. Morgan for affine resolvable designs, and determine the uniform (n × n)∕k semi-Latin squares with minimum PV aberration when there exist n−1 mutually orthogonal Latin squares of order n. These do not exist when n=6, and the smallest uniform semi-Latin squares in this case have size (6 × 6)∕10. We present a complete classification of the uniform (6 × 6)∕10 semi-Latin squares, and display the one with least PV aberration. We give a construction producing a uniform ((n + 1) × (n + 1)) ∕ ((n − 2)n) semi-Latin square when there exist n − 1 mutually orthogonal Latin squares of order n, and determine the PV aberration of such a uniform semi-Latin square. Finally, we describe how certain affine resolvable designs and balanced incomplete-block designs can be constructed from uniform semi-Latin squares. From the uniform (6 × 6)∕10 semi-Latin squares we classified, we obtain (up to block design isomorphism) exactly 16875 affine resolvable designs for 72 treatments in 36 blocks of size 12 and 8615 balanced incomplete-block designs for 36 treatments in 84 blocks of size 6. In particular, this shows that there are at least 16875 pairwise non-isomorphic orthogonal arrays OA (72,6,6,2).
Bailey , R A & Soicher , L H 2021 , ' Uniform semi-Latin squares and their pairwise-variance aberrations ' , Journal of Statistical Planning and Inference , vol. 213 , pp. 282-291 . https://doi.org/10.1016/j.jspi.2020.12.003
Journal of Statistical Planning and Inference
Copyright © 2020 Elsevier B.V. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1016/j.jspi.2020.12.003.
DescriptionThe support from EPSRC grant EP/M022641/1 (CoDiMa: a Collaborative Computational Project in the area of Computational Discrete Mathematics) is gratefully acknowledged.
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