
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. 150168.
He describes a square array of blocks in which
the treatments are as ``equalized''as they are in a Latin square.

F. Yates:
Complex experiments.
Journal of the Royal Statistical Society, Supplement (forerunner
of Series B)
2
(1935),
pp. 181223.
In Section 6 he approves of Latin squares with
splitplots but deprecates other semiLatin squares because there is
no randomization validity for the analysis which ignores
blocks.

R. A. Fisher:
Contribution to the discussion of the above paper.
Journal of the Royal Statistical Society, Supplement
2
(1935),
pp. 229231.
Noted that an analysis that takes blocks into
account is OK, just like any other incompleteblock design.

B. Harshbarger and L. L. Davis:
Latinized rectangular lattices.
Biometrics
8
(1952),
pp. 7384.
Use in food industry. Two treatment factors, with
nk and n levels, a third factor with n
levels. Introduction of term Latinized for semiLatin square.

B. Rojas and R. F. White:
The modified Latin square.
Journal of the Royal Statistical Society, Series B
19
(1957),
pp. 305317.
Use in agricultural field trials.
Randomization investigation.

L. A. Darby and N. Gilbert:
The Trojan square.
Euphytica
7
(1958),
pp. 183188.
Use in glasshouse experiments. Introduction of
Trojan squares.

S. H. Y. Hung and N. S. Mendelsohn:
On Howell designs.
Journal of Combinatorial Theory, Series A
16
(1974),
pp. 174198.
Introduction of Howell designs.

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

L. D. Andersen and A. J. W. Hilton:
Generalized Latin rectangles. I: Construction and decomposition.
Discrete Mathematics
31
(1980),
pp. 125152.
1979.
Some constructions.

L. D. Andersen and A. J. W. Hilton:
Generalized Latin rectangles. II: Embedding.
Discrete Mathematics
31
(1980),
pp. 235260.
1979.
Some constructions.

D. A. Preece and G. H. Freeman:
SemiLatin squares and related designs.
Journal of the Royal Statistical Society, Series B
45
(1983)
pp. 267277.
Historical survey. Some
constructions. Enumeration of isomorphism classes of (4×4)/2 squares.
Conjecture that all simple semiLatin squares are Trojan.

E. F. Brickell:
A few results in message authentication.
Congressus Numerantium
43
(1984),
pp. 141154.
Introduction of orthogonal multiarrays. One
discovery of the dodecahedral semiLatin square and a Latin square
orthogonal to it.

R. A. Bailey:
Restricted randomization for neighbourbalanced designs.
Statistics and Decisions, Supplement
2
(1985),
pp. 237248.
Use in construction and randomization of
neighbourbalanced completeblock designs.

D. Rasch and G. Herrendörfer:
Experimental Design.
Reidel, Dordrecht,
1986.
Called pseudoLatin squares.

A. Rosa and D. R. Stinson:
Onefactorizations of regular graphs and Howell designs of small
order.
Utilitas Mathematica
29
(1986),
pp. 99124.
Definition of Howell cube.

E. R. Williams:
Row and column designs with contiguous replicates.
Australian Journal of Statistics
28
(1986),
pp. 154163.
Called Latinized incompleteblock
designs. Use in experiments on irrigated cotton.

R. A. Bailey and C. A. Rowley:
Valid randomization.
Proceedings of the Royal Society of London, Series A
410
(1987),
pp. 105124.
Use in construction and randomization of
neighbourbalanced completeblock designs.

E. Seah and D. R. Stinson:
An assortment of new Howell designs.
Utilitas Mathematica
31
(1987),
pp. 175188.
Found all (6 × 6)/2 Howell designs whose graph has an automorphism
group which is transitive on vertices.

R. A. Bailey:
SemiLatin squares.
Journal of Statistical Planning and Inference
18
(1988),
pp. 299312.
Some constructions. Counterexample 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=n1.

D. A. Preece:
SemiLatin squares.
In Encyclopedia of Statistical Sciences,
(ed. S. Kotz and N. L. Johnson),
6,
John Wiley & Sons, New York,
1988,
pp. 359361.

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,
SpringerVerlag, New York,
1990,
pp. 122139.
Used as doubly perfect authentication
schemes.

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. 107112.
Use in sugarbeet trials.

R. A. Bailey:
An efficient semiLatin square for twelve treatments in blocks of size
two.
Journal of Statistical Planning and Inference
26
(1990),
pp. 262266.
Another discovery of the dodecahedral
semiLatin square.

C.S. Cheng and R. A. Bailey:
Optimality of some twoassociateclass partially balanced
incompleteblock designs.
Annals of Statistics
19
(1991),
pp. 16671671.
Trojan squares are optimal even among
unresolved incompleteblock designs.

R. A. Bailey:
Efficient semiLatin squares.
Statistica Sinica
2
(1992),
pp. 413437.
Correction of mistakes in Bailey (1988).
Optimality of inflations of (n × n)/(n1) Trojan
squares. Good semiLatin squares with (n,k) = (4,4),
(6,2) and (6,3).

R. A. Bailey:
Recent advances in experimental design in agriculture.
Bulletin of the International Statistical Institute
55 (1)
(1993),
pp. 179193.
Section 2 is about semiLatin squares. Unresolved incompleteblock
designs may be better: it depends on whether information is combined.

R. R. Sitter:
Balanced repeated replications based on orthogonal multiarrays.
Biometrika
80,
(1993),
pp. 211221.
Generalization of OMA to BOMA. Use in stratified sampling.

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

N. C. K. Phillips and W. D. Wallis:
All solutions to a tournament problem.
Congressus Numerantium
114
(1996),
pp. 193196.
Introduced the acronym SOMA for a simple orthogonal multiarray.
A SOMA(k,n) is a (n×n)/k semilatin
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.

R. A. Bailey:
A Howell design admitting A_{5}.
Discrete Mathematics
167168
(1997),
pp. 6571.
More elegant derivation of the dodecahedral semiLatin squares with
n=6 and k= 2 or 3, and their automorphism groups.

R. A. Bailey and P. E. Chigbu:
Enumeration of semiLatin squares.
Discrete Mathematics
167168
(1997),
pp. 7384.
Regarded a (n × n)/k semiLatin square as a
collection of k permutations in S_{n}.
Strong and weak isomorphism.
Reported
the number of isomorphism classes for n=4 and k= 2 or 3
or 4.

R. A. Bailey and G. Royle:
Optimal semiLatin squares with side six and block size two.
Proceedings of the Royal Society, Series A
453
(1997),
pp. 19031914.
Optimal simple semiLatin squares with n=6 and k=2.

R. N. Edmondson:
Trojan square and incomplete Trojan square designs for crop research.
Journal of Agricultural Science
131
(1998),
pp. 135142.
Construction, use in experiments, analysis of
data, with real examples.

P. E. Chigbu:
The block structures and isomorphism
of semiLatin squares and related designs.
Bulletin of the Institute of Combinatorics
and its Applications
25,
(1999),
pp. 5365.
Explains what isomorphism of semiLatin squares
means. Distinguishes semiLatin squares from somewhat similar designs
with different block structures.

P. E. Chigbu:
Optimal semiLatin squares for sixteen treatments in blocks of size four.
Journal of the Nigerian Statistical Association
13
(1999),
pp. 1125.
The three optimal squares with n=k=4
(they have the same canonical efficiency factors).

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. 138152.
How to use semiLatin squares in
practice. Designs for 2^{3} factorial structure in a (4×4)/2
semiLatin square.

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
http://www.combinatorics.org.
Treats a semiLatin square as a collection of
permutations. Discovery of some simple semiLatin squares with
(n,k) = (10,3) and (14,4). Discussion of decomposability.
R. A. Bailey
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R. A. Bailey
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24/6/00