In this paper, a construction equivalent to "sum construction", of BIBDs where BIBD is added to a BIBD that is automorphic to it is presented. The result is that we can get new BIBDs by forming the collection of t-(ν*,k,*λ_{t}) BIBD with its automorphic BIBDs. A new recursive technique has been developed for the construction of designs. It has also clearly shown that every t-design is also a BIB (ν*,k,*λ_{t}) design. Therefore, this construction technique also generates BIBDs. Therefore, this note presents an alternative method that is simpler and unified for the construction of BIBDs that are very important in the experimental designs. As it provides designs for different values of k, unlike many methods that provide designs for a single value of k. Moreso, it provides both Steiner and non-Steiner designs.

Periodical:

Advanced Trends in Mathematics (Volume 1)

Pages:

1-14

Citation:

G.S. Duggal and N.B. Okelo, "Sum Construction of Automorphic Bibds and their Applications in Experimental Designs", Advanced Trends in Mathematics, Vol. 1, pp. 1-14, 2014

Online since:

Dec 2014

Authors:

Distribution:

Open Access

This work is licensed under a

Creative Commons Attribution 4.0 International License

References:

Adhikari, B. (1967). On the symmetric differences of pairs of blocks of incomplete block designs, Calcutta stat. Assoc. Bull 16, 45-48.

Anderson, I. (1989). A first Course in Combinatorial Mathematics second edition, Oxford University Press, New York.

Blanchard, J. L (1995a). A construction for Steiner 3-designs, Journal of combinatorial Theory A, 71, 60-67.

Blanchard, J.L. (1995b). An extension theorem for Steiner systems, Discrete Mathematics 141, no. 1-3, 23-35.

Blanchard, J. L (1995c). A construction for orthogonal arrays with strength t ≥ 3, Discrete math 137, no. 1-3, 35-44.

Bogart, K.P. (1990). Introductory Combinatorics second edition, Harcourt Brace Jovanovich, Inc. Orlando, Florida.

Bose, R.C. (1950). A note on orthogonal arrays, Annals of math. stat. 21, 304-305.

Cameron, P. J., Maimani, H. R., Omidi, G.R., and Tayfeh-Rezaie, B. (2006). 3-designs PGL (2, q), Discrete Mathematics, 306, vol. 23, 3063-3073.

Colbourn, C.J. et al. (2002). Orthogonal arrays of strength three from regular 3-wise balanced designs.

Chowla, S and Ryser H.J. (1950). Combinatorial problems. Canadian journal of mathematics. 2, 93-9.

Dinitz, J. H., & Stinson, D. R. (1992). Contemporary Design Theory: A collection of surveys, Wiley-Interscience.

Fisher R.A. (1940). An examination of the difference possible solutions of a problem in incomplete blocks. Annals of Engenics 10, 52-75.

Hanan, H. (1960). On quadruple systems, Canadian Journal of mathematics, 15, 145-157.

Hanani, H. (1963). On some tactical configurations, Canadian Journal of mathematics, 15, 702722.