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Non-Existence of Some Unsymmetrical Partially Balanced Incomplete Block Designs

Published online by Cambridge University Press:  20 November 2018

S. S. Shrikhande
Affiliation:
Banaras Hindu University and Bombay University
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A partially balanced incomplete block (PBIB) design with m-associate classes is defined by Bose and Shimamoto (4) as follows:

(i) The experimental material is divided into b blocks of k units each, different treatments being applied to the units in the same block.

(ii) There are v treatments each of which occurs in r blocks.

(iii) There can be established a relation of association between any two treatments satisfying the following requirements.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Bose, R. C. and Connor, W. S., Combinatorial properties of group divisible incomplete block designs, Ann. Math. Stat., 23 (1952), 367383.Google Scholar
2. Bose, R. C. and Nair, K. R., Partially balanced incomplete block designs, Sankhyâ, 4 (1939), 337372.Google Scholar
3. Bose, R. C. and Dale M. Mesner, On linear associative algebras corresponding to association schemes of partially balanced designs, Ann. Math. Stat. 30 (1959), 2138.Google Scholar
4. Bose, R. C. and Shimamoto, T., Classification and analysis of partially balanced incomplete block designs with two associate classes, J. Amer. Stat. Assoc, 47 (1952), 151184.Google Scholar
5. Bruck, R. H. and Ryser, H. J., The nonexistence of certain finite projective planes, Can. J. Math., 1 (1949), 8893.Google Scholar
6. Connor, W. S. and Clatworthy, W. H., Some theorems for partially balanced designs, Ann. Math. Stat., 25 (1954), 100112.Google Scholar
7. A, L. C.. Corsten, Proper spaces related to triangular partially balanced incomplete block designs, Ann. Math. Stat., 31 (1960), 498501.Google Scholar
8. Jones, B. W., The arithmetic theory of quadratic forms (New York, 1952).Google Scholar
9. Laha, R. G. and Roy, J., Classification and analysis of linked block designs, Sankhyâ, 17 (1956-57), 115132.Google Scholar
10. Nair, K. R., Certain inequality relationships among the combinatorial parameters of incomplete block designs, Sankhyâ, 6 (1942-44), 255259.Google Scholar
11. Ogawa, Junjiro, A necessary condition for existence of regular symmetrical experimental designs of triangular type with partially balanced incomplete blocks, Ann. Math. Stat., 30 (1959), 10631071.Google Scholar
12. Pall, G., The arithmetical invariant of quadratic forms, Bull. Amer. Math. Soc, 51 (1945), 185197.Google Scholar
13. Raghavarao, Damaraju, A generalization of group divisible designs, Ann. Math. Stat., 31 (1960), 756771.Google Scholar
14. Saraf, W. S., On the structure and combinatorial properties of certain semi-regular group divisible designs, Sankhyâ, 23 (1961), Series A, 287296.Google Scholar
15. Shrikhande, S. S., The impossibility of certain symmetrical balanced incomplete block designs, Ann. Math. Stat., 21 (1950), 106111.Google Scholar
16. Shrikhande, S. S., On the dual of some balanced incomplete block designs, Biometrics, 8 (1952), 6672.Google Scholar
17. Shrikhande, S. S., The non-existence of certain affine resolvable balanced incomplete block designs, Can. J. Math., 5 (1953), 413420.Google Scholar
18. Shrikhande, S. S., Relations between certain incomplete block designs, Contributions to Probability and Statistics (1960), Stanford University Press, 388395.Google Scholar
19. Shrikhande, S. S. and Jain, N. C., The nonexistence of some partially balanced incomplete block designs with latin square type association schemes, Sankhyâ, 24 (1962), Ser. A, 259268.Google Scholar
20. Vartak, Manohar Narhar, The non-existence of certain PB IB designs, Ann. Math. Stat., 30 (1959), 10511062.Google Scholar