REFERENCES
REFERENCES
Abel R.J.R., Colbourn C.J., and Dinitz J.H., 2007. Mutually Orthogonal Latin Squares
(MOLS), chapter in The CRC Handbook of Combinatorial Designs, second edition,
edited by Charles J. Colbourn and J. H. Dinitz, Discrete Mathematics and its
Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton FL, pp. 160-193.
Addelman S., 1967. Equal and proportional frequency squares. J, Amer. Statist. Assoc.
62:226-240.
Aigner M. and Ziegler G.M., 1998. Proofs from the book. Chapter 27: Completing Latin
squares, third edition, Springer, pp. 179-184.
Aigner M. and Ziegler G.M., 1998. Proofs from the book. Chapter 23: Three famous
theorems on nite sets, third edition, Springer, pp. 151-155.
Andersen, Lars Dovling, 2007. The history of Latin Squares, Aalborg University
Denmark.
Bailey R.A., 1984. Quasi-complete Latin squares: Construction and randomization,
Journal of the Royal Statistical Society, B 46, 323-334.
Bailey R.A., 2008. "6 Row-Column designs and 9 More about Latin squares". Design of
Comparative
Experiments.
Cambridge
University
Press. ISBN 978-0-521-68357
9. MR 2422352. Pre-publication chapters are available on-line.
Balasubramanian K., 1990. On Transversals in Latin Squares, Linear Algebra Appl.,
Vol. 131, pp. 125-129.
Bammel S.E. and Rothstein J., 1975. The number of 9 _ 9 Latin squares, Discrete
Math., Vol. 11, N. 1, pp. 93-95.
128
Berger T., 1994. Classi cation des groupes de permutations d'un corps ni contenant le
groupe a ne, C. R. Acad. des Sciences de Paris, ser. I, pp. 117-119.
Bose R.C., 1938. On the application of properties of Galois fields to the problem of
construction of hypergraecolatin squares, Sankhya, 3, 328-338.
Bose R.C., and Shrikhande S.S., 1959(a). On the falsity of Euler’s conjecture about the
non-existence of two orthogonal Latin squares of order 4t+2. Proc. Natl. Acad. Sci. U.S.,
45, 734-737.
Bose R.C., and Shrikhande S.S., 1959(b). On the Construction of Sets of Pairwise
Orthogonal Latin Squares and the Falsity of a Conjecture of Euler. Mimeo Series No.
222, Institute of Statistics, University of North Carolina, Chapel Hill.
Bose, R.C., and Shrikhande S.S., 1959(c). On the Construction of Sets of Pairwise
Orthogonal Latin Squares and the Falsity of a Conjecture of Euler- II. Mimeo Series No.
225, Institute of Statistics, University of North Carolina, Chapel Hill.
Bose R.C., and Shrikhande S.S., 1960. On the composition of balanced incomplete block
designs. Can. J. Math., 12, 177-188.
Bose R.C., and Shrikhande S.S., Parker E.T., 1960. Further results on the construction of
mutually orthogonal latin squares and the falsity of Euler’s conjecture. Can. J. Math., 12,
189-203.
Bose R.C., https://en.wikipedia.org/wiki/Raj_Chandra_Bose.
Bush, K.A., 1952(a). A gemeralization of theorem due to MacNeish. Ann. Math. Stat., 23,
293-295.
Chowla S., Erdos P., and Straus E. G., 1960. On the maximal number of pairwise
orthogonal latin square of a given order. Can. J. Math., 12, 204-208.
Cochran W.G., Cox G.M., 1957. Experimental designs, 2nd Ed., Wiley, New York.
129
Colbourn C.J. and Dinitz J.H. (eds.), 2007. The CRC Handbook of Combinatorial
Designs, second edition, Discrete Mathematics and its Applications (Boca Raton),
Chapman & Hall/CRC, Boca Raton FL, xxii+984 pages.
Colbourn C.J., Dinitz J.H., and Wanless I.M., 2007. Latin Squares, chapter in The CRC
Handbook of Combinatorial Designs, second edition, edited by Charles J. Colbourn and
Je_rey H. Dinitz, Discrete Mathematics and its Applications (Boca Raton), Chapman &
Hall/CRC, Boca Raton FL, pp. 135-152.
Damaraju Raghavarao, 1988. Constructions and Combinatorial Problems in Design of
Experiments (corrected reprint of the 1971 John Wiley Series in Probability and
mathematical statistics). New York: Dover. ISBN 0-486-65685-3. MR 1102899.
Das M.N. and Giri N.C., 1986. Design and analysis of experiments. Wiley Eastern
Limited, New Delhi.
Dénes J., Keedwell A. D., 1974. Latin squares and their applications. New YorkLondon: Academic Press. p. 547. ISBN 0-12-209350-X. MR 0351850.
Dénes J., Keedwell A. D., 1991. Latin squares: New developments in the theory and
applications. Annals of Discrete Mathematics. 46. Paul Erdős (foreword). Amsterdam:
Academic Press. pp. xiv+454. ISBN 0-444-88899-3. MR 1096296.
Dénes J., Keedwell A.D., 1974. Latin squares and their applications, English
Universities, London.
Dey A., 1986. Theory of block designs. Wiley Eastern Ltd., New Delhi.
Dieter Jungnickel, 2001. The Geometry of Frequency Squares. Journal of Combinatorial
Theory. Series A 96 376-387.
130
Dr Nada Lakić, 2002. Classification of Latin Squares. Journal of Agricultural Sciences,
Vol. 47, No. 1, pg 105-112.
Euler L., 1782. Recherches sur une nouvelle esp_ece de quarr_es magiques, Verh.
Zeeuwsch. Genootsch. Wetensch. Vlissingen, Vol. 9, pp. 85-239.
Euler L., 1849. De quadratis magicis, Commentationes Arithmeticae, Vol. 2, pp. 593602.
Felgenhauer B. and Jarvis F, June 20, 2005. Enumerating possible Sudoku grids,
http://www.afjarvis.sta_.shef.ac.uk/sudoku/sudoku.pdf.
Finney D. J., 1945. Same orthogonal properties of the 4 X 4 and 6 X 6 Latin squares.
Ann. Eugen. 12:213-219.
Finney D. J., 1946. Orthogonal partitions of the 5 X 5 Latin squares. Ann. Eugen. 13:1-3.
Finney D. J., 1946. Orthogonal partitions of the 6 X 6 Latin squares. Ann. Eugen. 13:184196.
Fisher R.A. and Yates, F., 1963. Statistical tables for biological, agricultural and medical
research. 6th edition, Longman Group Ltd., London.
Freeman G. H., 1966. Some non-orthogonal partition of 4 X 4, 5 X 5, and 6 X 6 Latin
squares. Ann. Math. Statist. 37:661-681.
Freeman G.H., 1979. Complete Latin squares and related experimental designs, Journal
of the Royal Statistical Society, B 41, 253-262.
Frolov M., 1890. Recherches sur les permutations carr_ees, J. Math. Sp_eciales, Vol. (3)
4, pp. 8-11, 25-30.
131
Ghosh I.S. and C. R. Rao. Design and analysis of experiments. Handbook of
Statistics. 13. Amsterdam: North-Holland Publishing Co. pp. 903–937. ISBN 0-44482061-2. MR 1492586.
Gill P.S., Shukla G.K., 1985. Experimental designs and their efficiencies for spatially
correlated observations in two dimensions, Communications in Statistics - Theory and
Methods, 14, 2181-2197.
Gronau H.D. O. F. and Droebes S., April 15, 2011. Orthogonal latin squares of Sudoku
type,http://www.algorithm.unibayreuth.de/en/research/alcoma2010/participants/talkgrona
u.pdf, Thurnau.
Harshbarger B., 1947. Rectangular lattices. Va. Agri. Expt. Stat. Memoir, 1, 1-26.
Hedayat A. and Federer W. T., 1969. An application of group theory to the existence and
non-existence of orthogonal latin squares. Biometrika, 56(3): to appear.
Hedayat A., 1969. On the theory of the existence, non-existence, and the construction of
mutually orthogonal F-squares and Latin squares. Ph.D. thesis, presented to the Graduate
School of Cornell University for partial fulfillment of author's Ph.D. degree in statistics.
Hedayat A., Seiden E., 1970. F-square and orthogonal F-squares design: A generalization
of Latin squares and orthogonal Latin square design, The Annals of Mathematical
Statistics, Vol. 41, 2035-2144.
Hicks K., Kaisen J., and Mullen G.L., Numbers of Latin squares of prime power orders
with orthogonal mates, http://cage.ugent.be/fq10/data/abstracts.pdf, p. 48, contributed
talk at the 10th International Conference on Finite Fields and their Applications, Ghent.
Hinkelmann Klaus, Kempthorne Oscar, 2005. Design and Analysis of Experiments,
Volume 2: Advanced Experimental Design (First Ed.). Wiley. ISBN 978-0-471-55177
5. MR 2129060.
132
Hinkelmann Klaus, Kempthorne Oscar, 2008. Design and Analysis of Experiments. I, II
(Second Ed.). Wiley. ISBN 978-0-470-38551-7. MR 2363107.
Hinkelmann Klaus, Kempthorne Oscar, 2008. Design and Analysis of Experiments,
Volume I: Introduction to Experimental Design (Second Ed.). Wiley. ISBN 978-0-47172756 9. MR 2363107.
Hulpke A., Kaski P., and Osterg P.R.J., 2011. The number of Latin squares of order 11,
Math. Comp. 80, pp. 1197-1219.
J. H. van Lint, R. M. Wilson, 1992. A Course in Combinatorics, Cambridge University
Press ISBN 0-521-42260-4.
Jarvis F., February 2, 2008. http://www.afjarvis.sta_.shef.ac.uk/sudoku.
Jones B. and Kenward M.G., 1989. Design and analysis of crossover trials. Chapman and Hall,
London. Kuehl, R.O. 2000. Design of experiments: statistical principles in research design and
analysis. Duxbury Press, Pacific Grove, CA.
Knuth, Donald, 2011. Volume 4A: Combinatorial Algorithms, Part 1. The Art of
Computer
Programming (First
ed.).
Reading,
Massachusetts:
Addison-Wesley.
pp. xv+883pp. ISBN 0-201-03804-8.
Kudrle J.M. and Menard S.B., 2007. Magic Squares, chapter in The CRC Handbook of
Combinatorial Designs, second edition, edited by Charles J. Colbourn and Je_rey H.
Dinitz, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC,
Boca Raton FL, pp. 524-528.
Laywine C.F. and Mullen G.L., 1998. Discrete Mathematics Using Latin Squares, WileyInterscience.
133
Laywine, Charles F., Mullen, Gary L., 1998. Discrete mathematics using Latin squares.
Wiley-Interscience Series in Discrete Mathematics and Optimization. New York: John
Wiley & Sons, Inc. pp. xviii+305. ISBN 0-471-24064-8. MR 1644242.
Lei Gao, 2005. Latin Squares in Experimental Design. Michigan State University.
Liebeck M.W., Praeger C.E., and Saxl J., 1987. A classi_cation of the maximal
subgroups of the _nite alternating and symmetric groups, Journal of Algebra, Vol. 111,
pp. 365-383.
MacNeish H.F., 1952. Eulers squares. Ann. Math., 23, 221-227.
Mann H.B., 1942. The construction of orthogonal latin squares. Ann. Math. Stat., 13,
418-423.
Mann H. B., 1944. On orthogonal Latin squares, Amer. Math. Soc. Bull. 50:249-257.
McGuire G., Tugemann B., and Civario G., 2012. There is no 16-Clue Sudoku: Solving
the Sudoku Minimum Number of Clues Problem, arXiv.org.
McKay B.D., Meynert A., and Myrvold W., 2007. Small Latin Squares, Quasigroups and
Loops, J. Combination Design, Vol. 15, pp. 98-119.
McKay B.D., and Rogoyski E., 1995. Latin squares of order ten, Electronic J. Combin.,
Vol. 2, N. 3.
McKay B.D., and Wanless I.M., 2005. On the number of Latin squares, Ann. Comb. 9,
pp. 335-344.
Mead R., Curnow R.N., and Hasted A.M., 2003. Statistical methods in agriculture and
experimental biology. Chapman and Hall/CRC, Boca Raton, FL. Williams, E.J. 1949.
134
Michael
L.,
1993.
Early
Chinese
Texts.
A
Bibliographical
Guide,
http://www.scribd.com/doc/50889127/Loewe-Michael-Ed-Early-Chinese-Textsa
Bibliographical-Guide-1993, The Society for the Study of Early China.
Parker E.T., 1958. Construction of some sets of pairwise orthogonal latin squares. Am.
Math. Soc. Notices 5, 815 (abstract).
Parker E.T., 1959. Construction of some sets of mutually orthogonal latin squares. Proc.
Am. Math. Soc., 10, 946-951.
Petersen K.F., The New Sudoku Players' Forum, http://forum.enjoysudoku.com/su-dokus-maths-t44-420.html#p11874.
Peterson J., 1901-1902. Les 36 officieurs. Am. Math., 413-417.
Preece D.A., Freeman G.H., 1983. Semi-Latin squares and related designs, Journal of the
Royal Statistical Society, B 45, 267-277.
Rojas B., White R.F., 1957. The modified Latin square, Journal of the Royal Statistical
Society, B 19, 305-317.
Royle G.F., A collection of 49151 distinct Sudoku con_gurations with 17 entries,
http://mapleta.maths.uwa.edu.au/_gordon/sudokumin.php.
Sade A., 1948. Enum_eration des carr_es latins, application au 7e ordre, conjecture pour
les ordres sup_erieurs, Published by the author, Marseille, 8 pp.
Scheffe H., 1959. The Analysis of Variance. Wiley, New York.
Shah Kirti R., Sinha Bikas K., 1989. "4 Row-Column Designs". Theory of Optimal
Designs. Lecture Notes in Statistics. 54. Springer-Verlag. pp. 66–84. ISBN 0-387-969918. MR 1016151.
135
Shah Kirti R., Sinha Bikas K., 1996. "Row-column designs". Theory of Optimal
Designs. Lecture Notes in Statistics.
Shah S.M. and Kage D.G. An Introduction to Constuction and Analysis Statistical
Designs. Queen’s Papers in Pure and Applied Mathematics 64.
Shrikhande S.S., https://en.wikipedia.org/wiki/Sharadchandra_Shankar_Shrikhande.
Shrikhande S.S., 1965. On a class of partially balanced incomplete block designs. Math.
Statist., 36, 1807-1814.
Small C., 1988. Magic squares over fields, American Mathematical Monthly, 7, 621-625.
Storme L., Finite Geometry, 2007. Chapter in The CRC Handbook of Combinatorial
Designs, second edition, edited by Charles J. Colbourn and J. H. Dinitz, Discrete
Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton FL,
pp. 702-728.
Street Anne Penfold, Street Deborah J., 1987. Combinatorics of Experimental Design.
New
York:
Oxford
University
Press.
pp. 400+xiv
pp. ISBN 0-19-853256-
3. MR 0908490. ISBN 0-19-853255-5.
Swaney M., on the History of Magic Squares, http://www.ismaili.net/mirrors/Ikhwan
08/magic squares.html.
Tarry, G., 1900. Le problem de 36 officieurs. Compt. Rend. Assoc. Franc. Av. Sci., 1,
122-123.
Tomar, J.S., Jaggi, Seema and Varghese, Cini (2005). On totally balanced block designs
for competition effects. Jour. Applied Statistics, 32(1), 87-97.
Typke R., http://rainer.typke.org/typo3temp/GB/3d5496c501.png.
Vanpoucke Jordy, 2012. Mutually Orthogonal Latin Squares and their generalizations.
Master thesis, Ghent University.
136
Walter T. Federer, 1975. On the existence and construction of a complete set of
orthogonal F (4t; 2t, 2t) squares, Cornell University.
Wanless I., Transversals in Latin Squares, http://www.karlin.m_.cuni.cz/_loops07/
presentations/wanless.pdf
Wells M. B., 1967. The number of latin squares of order 8, J. Combin. Theory, Vol. 3,
pp. 98-99.
Youden W.J., Hunter J.S., 1955. Partialy replicated Latin squares, Biometrics,11,399405.
(Not all references are cited in the thesis)
137
List of Publications
1. A short history of Euler Conjecture and its Disproval published in
International Journal of Research and Analytical Reviews
(Volume 5; Page no. 339-340).
E ISSN 2348-1269; Print ISSN 2349-5138.
2. Simplified Method of construction of a complete set of MOLS.
(Submitted)
3. Simplified Method of construction of L1 of a complete set of MOLS.
(Submitted)
4. On frequency squares.
(Submitted)
138