DMPS

ISSN 1509-9423 (print version)

ISSN 2084-0381 (electronic version)

https://doi.org/10.7151/dmps

Discussiones Mathematicae Probability and Statistics

Discussiones Mathematicae Probability and Statistics

Article in volume

PDF

Discussiones Mathematicae Probability and Statistics 24(1) (2004) 41-58
DOI: https://doi.org/10.7151/dmps.1045

ON A CHARACTERIZATION OF SYMMETRIC BALANCED INCOMPLETE BLOCK DESIGNS

R.N.  Mohan,

P.G. Department of Mathematics, Sir C.R.R. College
Eluru-534007, India
e-mail: vjwrnmohan@sancharnet.in

Sanpei Kageyama

Hiroshima University,
Higashi-Hiroshima 739-8524, Japan
e-mail: ksanpei@hiroshima-u.ac.jp

M.M. Nair

Sarathi Institute of Engineering and Technology
Nuzvid-521201, India

Abstract

All the symmetric balanced incomplete block (SBIB) designs have been characterized and a new generalized expression on parameters of SBIB designs has been obtained. The parameter b has been formulated in a different way which is denoted by bi, i = 1, 2, 3, associating with the types of the SBIB design Di. The parameters of all the designs obtained through this representation have been tabulated while corresponding them with the suitable formulae for the number ofblocks bi and the expression Si for the convenience of practical users for constructional methods of certain designs, which is the main theme of this paper.

Keywords: symmetric balanced incomplete block (SBIB) design.

2000 Mathematics Subject Classification: 05B05.

References

[1] S. Chowla and H.J. Ryser, Combinatorial problems, Can. J. Math. 2 (1950), 93-99.
[2]R.J. Collins, Constructing BIB designs with computer, Ars Combin. 2 (1976), 285-303.
[3]J.D. Fanning, A family of symmetric designs, Discrete Math. 146 (1995), 307-312.
[4] Q.M. Hussain, Symmetrical incomplete block designs with l = 2,k = 8 or 9, Bull. Calcutta Math. Soc. 37 (1945), 115-123.
[5]Y.J. Ionin, A technique for constructing symmetric designs, Designs, Codes and Cryptography 14 (1998), 147-158.
[6]Y.J. Ionin, Building symmetric designs with building sets, Designs, Codes and Cryptography 17 (1999), 159-175.
[7]S. Kageyama, Note on Takeuchi's table of difference sets generating balanced incomplete block designs, Int. Stat. Rev. 40, (1972), 275-276.
[8]S. Kageyama and R.N. Mohan, On m-resolvable BIB designs, Discrete Math. 45 (1983), 113-121.
[9]R. Mathon and A. Rosa, 2-(v, k, l) designs of small order, The CRC Handbook of Combinatorial Designs (ed. Colbourn, C. J. and Dinitz, J. H.). CRC Press, New York, (1996), 3-41.
[10]R.N. Mohan, A note on the construction of certain BIB designs, Discrete Math. 29 (1980), 209-211.
[11]R.N. Mohan, On an Mn-matrix, Informes de Matematica (IMPA-Preprint), Series B-104, Junho/96, Instituto de Matematica Pura E Aplicada, Rio de Janeiro, Brazil 1996.
[12]R.N. Mohan, A new series of affine m-resolvable (d+1)-associate class PBIB designs, Indian J. Pure and Appl. Math. 30 (1999), 106-111.
[13]D. Raghavarao, Constructions and Combinatorial Problems in Design of Experiments, Wiley, New York 1971.
[14]S.S. Shrikhande, The impossibility of certain symmetrical balanced incomplete designs, Ann. Math. Statist. 21 (1950), 106-111.
[15]S.S. Shrikhande and N.K. Singh, On a method of constructing symmetrical balanced incomplete block designs, Sankhy¯a A24 (1962), 25-32.
[16]S.S. Shrikhande and D. Raghavarao, A method of construction of incomplete block designs, Sankhy¯a A25 (1963), 399-402.
[17]G. Szekers, A new class of symmetrical block designs, J. Combin. Theory 6 (1969), 219-221.
[18]K. Takeuchi, A table of difference sets generating balanced incomplete block designs, Rev. Inst. Internat. Statist. 30 (1962), 361-366.
[19] N.H. Zaidi, Symmetrical balanced incomplete block designs with l = 2 and k = 9, Bull. Calcutta Math. Soc. 55 (1963), 163-167.

Received 15 September 2003

Close