zbMATH — the first resource for mathematics

Construction of balanced incomplete block designs using cyclic shifts. (English) Zbl 1321.05020
Among numerous experimental designs the most comfortable and congenial are the combinatorial designs and the balanced incomplete block designs of them are highly practicable. And so the developing of several constructional methods of these BIB designs became a compulsion. This field of combinatorial designs was originated by Fisher and Yates and much developed by R. C. Bose and further extended to many ramifications in the field. These BIB designs are defined as: \(v\) elements are grouped into \(b\) blocks of equal size \(k\) such that each element occurs for \(r\) times and each pair occurs for \(\lambda\) times. The parametric relations satisfying these relations can be seen in the literature. Constructional methods were given by many authors like Bose, Shrikhande, Raghavarao, and so many others as shown in the paper. In the present paper the authors produce some infinite series of these designs by the method of cyclic shifts and hence it is very interesting. The authors ingeniously developed eleven such distinct infinite series of these designs.
05B05 Combinatorial aspects of block designs
62K10 Statistical block designs
Full Text: DOI
[1] Ahmed R., Pakistan Journal of Commerce and Social Sciences 5 (1) pp 100– (2011)
[2] DOI: 10.1111/j.1469-1809.1939.tb02219.x · Zbl 0023.00102 · doi:10.1111/j.1469-1809.1939.tb02219.x
[3] Bose R.C., Bulletin of the Calcutta Mathematical Society 34 pp 17– (1942)
[4] Bose R.C., Sankhya 6 pp 1– (1942)
[5] Bose R.C., Sankhya 8 pp 249– (1947)
[6] DOI: 10.1214/aoms/1177729958 · Zbl 0034.23102 · doi:10.1214/aoms/1177729958
[7] Cochran W.G., Experimental Designs. 2nd ed. (1957) · Zbl 0077.13205
[8] Davies O.L., Design and Analysis of Industrial Experiments. 2nd ed. (1956)
[9] DOI: 10.1002/bimj.4710370802 · Zbl 0850.62607 · doi:10.1002/bimj.4710370802
[10] Fisher R.A., Statistical Tables for Biological, Agricultural and Medical Research (1938) · JFM 64.1202.02
[11] Fisher R.A., Statistical Tables. 3rd ed. (1948)
[12] Hall M., Combinatorial Theory (1967)
[13] Iqbal I., Construction of Experimental Design Using Cyclic Shifts (1991)
[14] DOI: 10.1007/s11425-009-0063-1 · Zbl 1177.05019 · doi:10.1007/s11425-009-0063-1
[15] DOI: 10.1002/bimj.4710230203 · Zbl 0477.62064 · doi:10.1002/bimj.4710230203
[16] Kiefer J., A Survey of Statistical Design and Linear Models pp 333– (1975)
[17] Montgomery D.C., Design and Analysis of Experiments (1976)
[18] DOI: 10.1093/biomet/63.1.83 · Zbl 0338.62044 · doi:10.1093/biomet/63.1.83
[19] DOI: 10.1002/bimj.4710270109 · doi:10.1002/bimj.4710270109
[20] Quinoulli M.H., The Design and Analysis of Experiments (1953)
[21] Raghavarao D., Constructions and Combinatorial Problems in Design of Experiments (1971) · Zbl 0222.62036
[22] DOI: 10.1080/01621459.1947.10501947 · doi:10.1080/01621459.1947.10501947
[23] Rao C.R., Sankhya A 23 pp 283– (1961)
[24] Selden E., Sankhya 25 pp 393– (1963)
[25] DOI: 10.1093/biomet/44.1-2.278 · Zbl 0077.33702 · doi:10.1093/biomet/44.1-2.278
[26] DOI: 10.4153/CJM-1954-032-6 · Zbl 0055.37704 · doi:10.4153/CJM-1954-032-6
[27] Sprott D.A., Sankhya 17 pp 185– (1956)
[28] Sprott D.A., Sankhya Series A 24 pp 203– (1962)
[29] Street, A.P., Street, D.J. (1987).Combinatorics of Experimental Design. New York/Oxford: Oxford University Press/Clarendon, pp. 400. · Zbl 0622.05001
[30] DOI: 10.1214/aoms/1177728192 · Zbl 0072.36701 · doi:10.1214/aoms/1177728192
[31] DOI: 10.1111/j.1469-1809.1936.tb02134.x · doi:10.1111/j.1469-1809.1936.tb02134.x
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.