Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T05:04:55.741Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Composition of Balanced Incomplete Block Designs

Published online by Cambridge University Press:  20 November 2018

R. C. Bose  and
S. S. Shrikhande
R. C. Bose
Affiliation:
University of North Carolina and Case Institute of Technology
S. S. Shrikhande
Affiliation:
University of North Carolina and Case Institute of Technology
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The object of this paper is to develop a method of constructing balanced incomplete block designs. It consists in utilizing the existence of two balanced incomplete block designs to obtain another such design by what may be called the method of composition.

1. Preliminary results on orthogonal arrays and balanced incomplete block designs. Consider a matrix A = (aij) of k rows and N columns, where each aij represents one of the integers 1, 2, … , s. Consider all t-rowed submatrices of N columns, which can be formed from this array, t ≤ k. Each column of any Crowed submatrix can be regarded as an ordered t-plet.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 177 - 188
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Bose, R.C., On the construction of balanced incomplete block designs, Ann. Eugen., 9 (1939), 353–399.Google Scholar
2. Bose, R.C. and Bush, K.A., Orthogonal arrays of strength two and three, Ann. Math. Stat., 23 (1952), 508–524.Google Scholar
3. Bose, R.C. and Shrikhande, S.S., On the falsity of Euler's conjecture about the nonexistence of two orthogonal Latin squares of order 4t + 2, Proc. N. A. S., 45 (1959), 734–737.Google Scholar
4. Bose, R.C. and Shrikhande, S.S., On the construction of sets of pairwise orthogonal Latin squares and the falsity of a conjecture of Ruler, Trans. Amer. Math. Soc, to appear.Google Scholar
5. Bose, R.C., Shrikhande, S.S., and Parker, E.T., Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler's conjecture, Can. J. Math., 12 (1960), 189–203.Google Scholar
6. Bush, K.A., Orthogonal arrays of index unity, Ann. Math. Stat., 28 (1952), 426–434.Google Scholar
7. Fisher, R.A., An examination of the different possible solutions of a problem in incomplete blocks, Ann. Eugen. London, 10 (1940), 52–75.Google Scholar
8. Hartley, H.O., Shrikhande, S.S., and Taylor, W.B., A note on incomplete block designs with row balance, Ann. Math. Stat., 24 (1953), 123–128.Google Scholar
9. Jacobi, C.G., Canon Arithmeticus, Akademie-verlag (Berlin, 1956).Google Scholar
10. MacNeish, H.F., Euler's squares, Ann. Math., Series 2, 23 (1922), 221–227.Google Scholar
11. Mann, H.B., Analysis and design of experiments (Dover, 1949).Google Scholar
12. Netto, E., Lehrbuch der combinatorik (Chelsea Publishing Co., New York).Google Scholar
13. Parker, E.T., Construction of some sets of mutually orthogonal Latin squares, Proc. Amer. Math. Soc, to appear.Google Scholar
14. Parker, E.T., Orthogonal Latin squares, Proc. N. A. S., 45 (1959), 859–862.Google Scholar
15. Plackett, R.L. and Burman, J.P., The design of optimum multi-factorial experiments, Biometrika, 33 (1943-6), 305–325.Google Scholar
16. Rao, C.R., Factorial experiments derivable from combinatorial arrangements of arrays, J. Royal Stat. Soc, Suppl. 9 (1947), 128–139.Google Scholar
17. Smith, C.A.B. and Hartley, H.O., The construction of Youden squares, J. Royal Stat. Soc, Series B, 10 (1948), 262–263.Google Scholar
18. Yates, F., Incomplete randomised blocks, Ann. Eugen. London, 8 (1936), 121–140.Google Scholar