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dc.contributor.authorBailey, R.A.
dc.contributor.authorSoicher, Leonard H.
dc.date.accessioned2021-12-19T00:41:23Z
dc.date.available2021-12-19T00:41:23Z
dc.date.issued2021-07
dc.identifier271800775
dc.identifiercad00ade-154a-4765-a954-6a5a6a24483c
dc.identifier85099192875
dc.identifier000617031900019
dc.identifier.citationBailey , 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.2020.12.003en
dc.identifier.issn0378-3758
dc.identifier.otherRIS: urn:B43C69CDFFA8AB8596BCC2CD002842DE
dc.identifier.otherORCID: /0000-0002-8990-2099/work/85855316
dc.identifier.urihttps://hdl.handle.net/10023/24539
dc.descriptionThe support from EPSRC grant EP/M022641/1 (CoDiMa: a Collaborative Computational Project in the area of Computational Discrete Mathematics) is gratefully acknowledged.en
dc.description.abstractFor 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).
dc.format.extent291079
dc.language.isoeng
dc.relation.ispartofJournal of Statistical Planning and Inferenceen
dc.subjectDesign optimalityen
dc.subjectBlock designen
dc.subjectSchur optimalityen
dc.subjectAffine resolvable designen
dc.subjectBalanced incomplete-block designen
dc.subjectOrthogonal arrayen
dc.subjectQA Mathematicsen
dc.subjectT-NDASen
dc.subject.lccQAen
dc.titleUniform semi-Latin squares and their pairwise-variance aberrationsen
dc.typeJournal articleen
dc.contributor.sponsorEPSRCen
dc.contributor.institutionUniversity of St Andrews. Statisticsen
dc.contributor.institutionUniversity of St Andrews. Centre for Interdisciplinary Research in Computational Algebraen
dc.identifier.doi10.1016/j.jspi.2020.12.003
dc.description.statusPeer revieweden
dc.date.embargoedUntil2021-12-19
dc.identifier.grantnumberEP/M022641/1en


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