×

Uniform semi-Latin squares and their pairwise-variance aberrations. (English) Zbl 1460.62121

Summary: For integers \(n > 2\) and \(k > 0\), an \(( n \times n ) / k\) semi-Latin square is an \(n \times n\) array of \(k\)-subsets (called blocks) of an \(n k\)-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 \(( n \times n ) / k\) semi-Latin square is Schur optimal in the class of all \(( n \times n ) / k\) semi-Latin squares, and here we show that when a uniform \(( n \times n ) / k\) semi-Latin square exists, the Schur optimal \(( n \times n ) / k\) 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 \(( n \times n ) / k\) 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 \(( 6 \times 6 ) / 10\). We present a complete classification of the uniform \(( 6 \times 6 ) / 10\) semi-Latin squares, and display the one with least PV aberration. We give a construction producing a uniform \(( ( n + 1 ) \times ( n + 1 ) ) / ( ( n - 2 ) n )\) 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 \(( 6 \times 6 ) / 10\) 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 \(\operatorname{OA} ( 72 , 6 , 6 , 2 )\).

MSC:

62K05 Optimal statistical designs
62K10 Statistical block designs
05B05 Combinatorial aspects of block designs
05B15 Orthogonal arrays, Latin squares, Room squares
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bailey, R. A., Efficient semi-Latin squares, Statist. Sinica, 2, 413-437 (1992) · Zbl 0822.62060
[2] Bailey, R. A., Variance and concurrence in block designs, and distance in the corresponding graphs, Michigan Math. J., 58, 105-124 (2009) · Zbl 1185.05016
[3] Bailey, R. A., Symmetric factorial designs in blocks, J. Stat. Theory Pract., 5, 13-24 (2011) · Zbl 05902632
[4] Bailey, R. A.; Cameron, P. J., A family of balanced incomplete-block designs with repeated blocks on which general linear groups act, J. Comb. Des., 15, 143-150 (2007) · Zbl 1112.05012
[5] Bailey, R. A.; Cameron, P. J., Combinatorics of optimal designs, (Huczynska, S.; etal., Surveys in Combinatorics 2009 (2009), Cambridge University Press: Cambridge University Press Cambridge), 19-74 · Zbl 1182.05010
[6] Bailey, R. A.; Monod, H.; Morgan, J. P., Construction and optimality of affine-resolvable designs, Biometrika, 82, 187-200 (1995) · Zbl 0828.62061
[7] Bose, R. C., On the application of the properties of Galois fields to the problem of construction of Hyper-Graeco-Latin squares, Sankhyā, 3, 323-338 (1938)
[8] Bose, R. C., A note on the resolvability of balanced incomplete block designs, Sankhyā, 6, 105-110 (1942) · Zbl 0060.31404
[9] Bulutoglu, D. A.; Margot, F., Classification of orthogonal arrays by integer programming, J. Statist. Plann. Inference, 138, 654-666 (2008) · Zbl 1139.62041
[10] Bulutoglu, D. A.; Ryan, K. J., Algorithms for finding generalized minimum aberration designs, J. Complexity, 31, 577-589 (2015) · Zbl 1320.62188
[11] Bulutoglu, D. A.; Ryan, K. J., Integer programming for classifying orthogonal arrays, Australas. J. Combin., 70, 362-385 (2018) · Zbl 1383.05031
[12] Caliński, T., On some desirable patterns in block designs, Biometrics, 27, 275-292 (1971)
[13] Ceranka, B.; Kageyama, S.; Mejza, S., A new class of C-designs, Sankhyā Ser. B, 48, 199-206 (1986) · Zbl 0625.62063
[14] Egan, J.; Wanless, I. M., Enumeration of MOLS of small order, Math. Comp., 85, 799-824 (2016) · Zbl 1332.05025
[15] Geyer, A. J.; Bulutoglu, D. A.; Ryan, K. J., Finding the symmetry group of an LP with equality constraints and its application to classifying orthogonal arrays, Discrete Optim., 32, 93-119 (2019) · Zbl 1474.90276
[16] Gibbons, P. B.; Östergård, P. R.J., Computational methods in design theory, (Colbourn, C. J.; Dinitz, J. H., Handbook of Combinatorial Designs (2007), Chapman and Hall/CRC: Chapman and Hall/CRC Boca Raton), 755-783
[17] Giovagnoli, A.; Wynn, H. P., Optimum continuous block designs, Proc. R. Soc. A, 377, 405-416 (1981) · Zbl 0463.05017
[18] Hedayat, A. S.; Sloane, N. J.A.; Stufken, J., Orthogonal Arrays (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0935.05001
[19] John, J. A.; Mitchell, T. J., Optimal incomplete block designs, J. R. Stat. Soc. Ser. B Stat. Methodol., 39, 39-43 (1977) · Zbl 0354.05015
[20] John, J. A.; Williams, E. R., Cyclic and Computer Generated Designs (1995), Chapman and Hall: Chapman and Hall London · Zbl 0865.05010
[21] Junttila, T.; Kaski, P., Engineering an efficient canonical labeling tool for large and sparse graphs, (Applegate, D.; etal., Proceedings of the Ninth Workshop on Algorithm Engineering and Experiments and the Fourth Workshop on Analytic Algorithmics and Combinatorics (2007), SIAM: SIAM Philadelphia), 135-149, bliss homepage: http://www.tcs.hut.fi/Software/bliss/ · Zbl 1428.68222
[22] Mathon, R.; Rosa, A., \(2- ( v , k , \lambda )\) designs of small order, (Colbourn, C. J.; Dinitz, J. H., Handbook of Combinatorial Designs (2007), Chapman and Hall/CRC: Chapman and Hall/CRC Boca Raton), 25-58 · Zbl 1113.05016
[23] Morgan, J. P., Optimal resolvable designs with minimum PV aberration, Statist. Sinica, 20, 715-732 (2010) · Zbl 1187.62133
[24] Owens, P. J.; Preece, D. A., Complete sets of pairwise orthogonal Latin squares of order 9, J. Combin. Math. Combin. Comput., 18, 83-96 (1995) · Zbl 0832.05013
[25] Rao, C. R., Factorial experiments derivable from combinatorial arrangements of arrays, Suppl. J. R. Stat. Soc., 9, 128-139 (1947) · Zbl 0031.06201
[26] Saha, G. M., On Calinski’s patterns in block designs, Sankhyā, Ser. B, 38, 383-392 (1976) · Zbl 0409.62062
[27] Schoen, E. D.; Eendebak, P. T.; Nguyen, M. V.M., Complete enumeration of pure-level and mixed-level orthogonal arrays, J. Comb. Des., 18, 123-140 (2010), and correction, 488 · Zbl 1287.05017
[28] Shah, K. R.; Sinha, B. K., Theory of Optimal Designs, Lecture Notes in Statistics (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0688.62043
[29] Soicher, L. H., Uniform semi-Latin squares and their Schur-optimality, J. Comb. Des., 20, 265-277 (2012) · Zbl 1248.05022
[30] Soicher, L. H., Designs, groups and computing, (Detinko, A.; etal., Probabilistic Group Theory, Combinatorics, and Computing. Lectures from the Fifth de Brún Workshop. Probabilistic Group Theory, Combinatorics, and Computing. Lectures from the Fifth de Brún Workshop, Lecture Notes in Mathematics, vol. 2070 (2013), Springer: Springer London), 83-107 · Zbl 1268.05029
[31] Soicher, L. H., Optimal and efficient semi-Latin squares, J. Statist. Plann. Inference, 143, 573-582 (2013) · Zbl 1428.62358
[32] Soicher, L. H., The DESIGN package for GAP, Version 1.7 (2019), https://gap-packages.github.io/design/
[33] Soicher, L. H., The GRAPE package for GAP, Version 4.8.3 (2019), https://gap-packages.github.io/grape/
[34] Suen, C.-Y., On construction of balanced factorial experiments (1982), Ph.D. thesis, University of North Carolina at Chapel Hill
[35] Suen, C.-Y.; Chakravarti, I. M., Efficient two-factor balanced designs, J. R. Stat. Soc. Ser. B Stat. Methodol., 48, 107-114 (1986) · Zbl 0592.62071
[36] The GAP Group, GAP — Groups, algorithms, and programming, Version 4.11.0 (2020), https://www.gap-system.org/
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.