Semi-Latin squares

Annotated Partial Bibliography of Semi-Latin squares

  1. W. S. Gosset (``Student''): Yield trials. Baillière's Encyclopedia of Scientific Agriculture. London, (1931), p. 1342, reprinted in ``Student's'' Collected Papers, edited by E. S. Pearson and J. Wishart, Biometrika, London, 1942, pp. 150-168.
    He describes a square array of blocks in which the treatments are as ``equalized''as they are in a Latin square.

  2. F. Yates: Complex experiments. Journal of the Royal Statistical Society, Supplement (fore-runner of Series B) 2 (1935), pp. 181-223.
    In Section 6 he approves of Latin squares with split-plots but deprecates other semi-Latin squares because there is no randomization validity for the analysis which ignores blocks.

  3. R. A. Fisher: Contribution to the discussion of the above paper. Journal of the Royal Statistical Society, Supplement 2 (1935), pp. 229-231.
    Noted that an analysis that takes blocks into account is OK, just like any other incomplete-block design.

  4. B. Harshbarger and L. L. Davis: Latinized rectangular lattices. Biometrics 8 (1952), pp. 73-84.
    Use in food industry. Two treatment factors, with nk and n levels, a third factor with n levels. Introduction of term Latinized for semi-Latin square.

  5. B. Rojas and R. F. White: The modified Latin square. Journal of the Royal Statistical Society, Series B 19 (1957), pp. 305-317.
    Use in agricultural field trials. Randomization investigation.

  6. L. A. Darby and N. Gilbert: The Trojan square. Euphytica 7 (1958), pp. 183-188.
    Use in glasshouse experiments. Introduction of Trojan squares.

  7. S. H. Y. Hung and N. S. Mendelsohn: On Howell designs. Journal of Combinatorial Theory, Series A 16 (1974), pp. 174-198.
    Introduction of Howell designs.

  8. L. D. Andersen: Latin squares and their generalizations. Ph.D. thesis, University of Reading, 1979.
    Some constructions.

  9. L. D. Andersen and A. J. W. Hilton: Generalized Latin rectangles. I: Construction and decomposition. Discrete Mathematics 31 (1980), pp. 125-152. 1979.
    Some constructions.

  10. L. D. Andersen and A. J. W. Hilton: Generalized Latin rectangles. II: Embedding. Discrete Mathematics 31 (1980), pp. 235-260. 1979.
    Some constructions.

  11. D. A. Preece and G. H. Freeman: Semi-Latin squares and related designs. Journal of the Royal Statistical Society, Series B 45 (1983) pp. 267-277.
    Historical survey. Some constructions. Enumeration of isomorphism classes of (4×4)/2 squares. Conjecture that all simple semi-Latin squares are Trojan.

  12. E. F. Brickell: A few results in message authentication. Congressus Numerantium 43 (1984), pp. 141-154.
    Introduction of orthogonal multi-arrays. One discovery of the dodecahedral semi-Latin square and a Latin square orthogonal to it.

  13. R. A. Bailey: Restricted randomization for neighbour-balanced designs. Statistics and Decisions, Supplement 2 (1985), pp. 237-248.
    Use in construction and randomization of neighbour-balanced complete-block designs.

  14. D. Rasch and G. Herrendörfer: Experimental Design. Reidel, Dordrecht, 1986.
    Called pseudo-Latin squares.

  15. A. Rosa and D. R. Stinson: One-factorizations of regular graphs and Howell designs of small order. Utilitas Mathematica 29 (1986), pp. 99-124.
    Definition of Howell cube.

  16. E. R. Williams: Row and column designs with contiguous replicates. Australian Journal of Statistics 28 (1986), pp. 154-163.
    Called Latinized incomplete-block designs. Use in experiments on irrigated cotton.

  17. R. A. Bailey and C. A. Rowley: Valid randomization. Proceedings of the Royal Society of London, Series A 410 (1987), pp. 105-124.
    Use in construction and randomization of neighbour-balanced complete-block designs.

  18. E. Seah and D. R. Stinson: An assortment of new Howell designs. Utilitas Mathematica 31 (1987), pp. 175-188.
    Found all (6 × 6)/2 Howell designs whose graph has an automorphism group which is transitive on vertices.

  19. R. A. Bailey: Semi-Latin squares. Journal of Statistical Planning and Inference 18 (1988), pp. 299-312.
    Some constructions. Counter-example to Preece and Freeman's conjecture. Randomization validity of the analysis which includes blocks. Efficiency factors for the quotient block design, including values in some cases (Lemma 3 and Theorem (b) are wrong!). Optimality of Trojan squares with k=n-1.

  20. D. A. Preece: Semi-Latin squares. In Encyclopedia of Statistical Sciences, (ed. S. Kotz and N. L. Johnson), 6, John Wiley & Sons, New York, 1988, pp. 359-361.

  21. M. G. H. Anthony, K. M. Martin, J. Seberry and P. Wild: Some remarks on authentication systems. Advances in Cryptology, Auscrypt '90, Lecture Notes in Computer Science 453, Springer-Verlag, New York, 1990, pp. 122-139.
    Used as doubly perfect authentication schemes.

  22. R. A. Bailey and R. W. Payne: Experimental design: statistical research and its application. In Institute of Arable Crops Research Report for 1989 (ed. J. Abbott), Agriculture and Food Research Council, Institute of Arable Crops Research, Harpenden, 1990, pp. 107-112.
    Use in sugar-beet trials.

  23. R. A. Bailey: An efficient semi-Latin square for twelve treatments in blocks of size two. Journal of Statistical Planning and Inference 26 (1990), pp. 262-266.
    Another discovery of the dodecahedral semi-Latin square.

  24. C.-S. Cheng and R. A. Bailey: Optimality of some two-associate-class partially balanced incomplete-block designs. Annals of Statistics 19 (1991), pp. 1667-1671.
    Trojan squares are optimal even among unresolved incomplete-block designs.

  25. R. A. Bailey: Efficient semi-Latin squares. Statistica Sinica 2 (1992), pp. 413-437.
    Correction of mistakes in Bailey (1988). Optimality of inflations of (n × n)/(n-1) Trojan squares. Good semi-Latin squares with (n,k) = (4,4), (6,2) and (6,3).

  26. R. A. Bailey: Recent advances in experimental design in agriculture. Bulletin of the International Statistical Institute 55 (1) (1993), pp. 179-193.
    Section 2 is about semi-Latin squares. Unresolved incomplete-block designs may be better: it depends on whether information is combined.

  27. R. R. Sitter: Balanced repeated replications based on orthogonal multi-arrays. Biometrika 80, (1993), pp. 211-221.
    Generalization of OMA to BOMA. Use in stratified sampling.

  28. P. E. Chigbu: Semi-Latin squares: methods for enumeration and comparison. Ph.D. thesis, University of London, 1995.
    Complete enumeration of isomorphism classes of semi-Latin squares with n=4 and k= 2 or 3 or 4. Optimal squares with n=k=4.

  29. N. C. K. Phillips and W. D. Wallis: All solutions to a tournament problem. Congressus Numerantium 114 (1996), pp. 193-196.
    Introduced the acronym SOMA for a simple orthogonal multi-array. A SOMA(k,n) is a (n×n)/k semi-latin square in which no pair of letters concur in a block more than once. Classification of all SOMA(3,6) and SOMA(4,6) up to strong isomorphism.

  30. R. A. Bailey: A Howell design admitting A5. Discrete Mathematics 167-168 (1997), pp. 65-71.
    More elegant derivation of the dodecahedral semi-Latin squares with n=6 and k= 2 or 3, and their automorphism groups.

  31. R. A. Bailey and P. E. Chigbu: Enumeration of semi-Latin squares. Discrete Mathematics 167-168 (1997), pp. 73-84.
    Regarded a (n × n)/k semi-Latin square as a collection of k permutations in Sn. Strong and weak isomorphism. Reported the number of isomorphism classes for n=4 and k= 2 or 3 or 4.

  32. R. A. Bailey and G. Royle: Optimal semi-Latin squares with side six and block size two. Proceedings of the Royal Society, Series A 453 (1997), pp. 1903-1914.
    Optimal simple semi-Latin squares with n=6 and k=2.

  33. R. N. Edmondson: Trojan square and incomplete Trojan square designs for crop research. Journal of Agricultural Science 131 (1998), pp. 135-142.
    Construction, use in experiments, analysis of data, with real examples.

  34. P. E. Chigbu: The block structures and isomorphism of semi-Latin squares and related designs. Bulletin of the Institute of Combinatorics and its Applications 25, (1999), pp. 53-65.
    Explains what isomorphism of semi-Latin squares means. Distinguishes semi-Latin squares from somewhat similar designs with different block structures.

  35. P. E. Chigbu: Optimal semi-Latin squares for sixteen treatments in blocks of size four. Journal of the Nigerian Statistical Association 13 (1999), pp. 11-25.
    The three optimal squares with n=k=4 (they have the same canonical efficiency factors).

  36. S. Ferris and S. G. Gilmour: Blocking factorial designs in greenhouse experiments. In Proceedings of the Tenth Annual Kansas State University Conference on Applied Statistics in Agriculture (1999), pp. 138-152.
    How to use semi-Latin squares in practice. Designs for 23 factorial structure in a (4×4)/2 semi-Latin square.

  37. Leonard H. Soicher: On the structure and classification of SOMAs: generalizations of mutually orthogonal Latin squares. Electronic Journal of Combinatorics, R32 of Volume 6(1), (1999), 15 pages, at
    Treats a semi-Latin square as a collection of permutations. Discovery of some simple semi-Latin squares with (n,k) = (10,3) and (14,4). Discussion of decomposability.

R. A. Bailey

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