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On the Composition of Balanced Incomplete Block Designs

  • R. C. Bose (a1) and S. S. Shrikhande (a1)

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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, tk. Each column of any Crowed submatrix can be regarded as an ordered t-plet.

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References

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1. Bose, R.C., On the construction of balanced incomplete block designs, Ann. Eugen., 9 (1939), 353399.
2. Bose, R.C. and Bush, K.A., Orthogonal arrays of strength two and three, Ann. Math. Stat., 23 (1952), 508524.
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), 734737.
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.
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), 189203.
6. Bush, K.A., Orthogonal arrays of index unity, Ann. Math. Stat., 28 (1952), 426434.
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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), 123128.
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18. Yates, F., Incomplete randomised blocks, Ann. Eugen. London, 8 (1936), 121140.
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On the Composition of Balanced Incomplete Block Designs

  • R. C. Bose (a1) and S. S. Shrikhande (a1)

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