Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T13:53:16.920Z Has data issue: false hasContentIssue false
  • English
  • Français

Construction methods for Bhaskar Rao and related designs

Part of: Designs and configurations

Published online by Cambridge University Press:  09 April 2009

Peter B. Gibbons  and
Rudolf Mathon
Peter B. Gibbons
Affiliation:
Department of Computer Science, University of Auckland, Auckland, New Zealand
Rudolf Mathon
Affiliation:
Department of Computer Science, University of Toronto, Toronto, Ontario, Canada, (M5S 1A4)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Mathématical and computational techniques are described for constructing and enumerating generalized Bhaskar Rao designs (GBRD's). In particular, these methods are applied to GBRD(k + 1, k, 1(k − 1); G)'s for 1 ≥ 1. Properties of the enumerated designs, such as automorphism groups, resolutions and contracted designs are tabulated. Also described are applications to group divisible designs, multi-dimensional Howell cubes, generalized Room squares, equidistant permutation arrays, and doubly resolvable two-fold triple systems.

MSC classification

Secondary: 05B05: Block designs 05B15: Orthogonal arrays, Latin squares, Room squares 05B25: Finite geometries
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 1 , February 1987 , pp. 5 - 30
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Rao, M. Bhaskar, ‘A group divisible family of PBIB designs, J. Indian Stat. Association 4 (1966), 14–28.Google Scholar
[2]Butson, A. T., ‘Generalized Hadamard matrices’, Proc. Amer. Math. Soc. 13 (1962), 894–898.CrossRefGoogle Scholar
[3]Butson, A. T., ‘Relations among generalized Hadamard matrices, relative difference sets and maximal lengh recurring sequences’, Canad. J. Math. 15 (1963), 42–48.CrossRefGoogle Scholar
[4]Colbourn, C. J. and Vanstone, S. A., ‘Doubly resolvable twofold triple systems’, Congressus Numerantium 34 (1982), 219–223.Google Scholar
[5]Delsarte, P. and Goethals, J. M., ‘On quadratic residue-like in Abelian groups’, Report R168, MBLE Research Laboratory, Brussels (1971).Google Scholar
[6]Delsarte, P. and Goethals, J. M., ‘Tri-weight codes and generalized Hadamard matrices’, Information and Control 15 (1969), 196–206.CrossRefGoogle Scholar
[7]Drake, D. A., ‘Partial λ-geometries and generalized Hadamard matrices over groups’, Canad. J. Math. 31 (1979), 617–627.CrossRefGoogle Scholar
[8]Geramita, A. V., Pullman, N. J. and Wallis, J. Seberry, ‘Families of weighing matrices’, Bull. Austral. Math. Soc. 10 (1974), 119–122.CrossRefGoogle Scholar
[9]Geramita, A. V. and Seberry, J., Orthogonal designs: quadratic forms and Hadamard matrices (Marcel Dekker, New York, 1979).Google Scholar
[10]Gibbons, P. B., ‘Computing techniques for the construction and analysis of block designs’, Tech. Report # 92, Dept. of Computer Science, University of Toronto (05 1976).Google Scholar
[11]Gibbons, P. B., Mathon, R. and Corneil, D. G., ‘Computing techniques for the construction and analysis of block designs’, Utilitas Math. 11 (1977), 161–192.Google Scholar
[12]Goethals, J. M. and Seidel, J. J., ‘Orthogonal matrices with zero diagonal’, Canad. J. Math. 19 (1967), 1001–1010.CrossRefGoogle Scholar
[13]Hain, R., Circulant weighing matrices (M. Sc. Thesis, Australian National University, Canberra 1977).Google Scholar
[14]Hollmann, H. D. L., Association schemes (M. Sc. thesis, Eindhoven University of Technology, 1982).Google Scholar
[15]Mathon, R., ‘Symmetric conference matrices of order pq 2 + 1’, Canad. J. Math. 30 (1978), 321–331.CrossRefGoogle Scholar
[16]Mathon, R., ‘3-class association schemes’ (in Proc. Conf. on Algebraic Aspects of Combinatorics, Toronto 1975, 123–155).Google Scholar
[17]Mathon, R. A., Phelps, K. T. and Rosa, A., ‘A class of Steiner triple systems of order 21 and associated Kirkman systems’, Math. of Comp. 37 (1981), 209–222.CrossRefGoogle Scholar
[18]Mathon, R. and Rosa, A., ‘The 4-rotational Steiner and Kirkman triple systems of order 21’, Ars Combinatoria 17 A (1984), 241–250.Google Scholar
[19]Mathon, R. and Vanstone, S. A., ‘On the existence of doubly resolvable Kirkman systems and equidistant permutation arrays’, Discrete Math. 30 (1980), 157–172.CrossRefGoogle Scholar
[20]Read, R. C., ‘Every one a winner’, Ann. Discrete Math. 2 (1978), 107–120.CrossRefGoogle Scholar
[21]Rosa, A., ‘Room squares generalized’, Ann. Discrete Math. 8 (1980), 43–57.CrossRefGoogle Scholar
[22]Seberrry, J., ‘Some families of partially balanced incomplete block designs’, (Combinatorial Mathematics IX, Lecture Notes in Math. 829, Springer, Berlin, 1982).Google Scholar
[23]Seberrry, J., ‘Some remarks on generalized Hadamard matrices and theorems of Rajkundlia on SBIBD's (Combinatorial Mathematics VI, Lecture Notes in Math. 748, Springer, 1979, 154–164).CrossRefGoogle Scholar
[24]Shrikhande, S. S., ‘Generalized Hadamard matrices and orthogonal arrays of strength 2’, Canad. J. Math. 16 (1964), 736–740.CrossRefGoogle Scholar
[25]Street, D. J. and Rodger, C. A., ‘Some results on Bhaskar Rao designs’, (Combinatorial Mathematics VII, edited by Robinson, R. W., Southern, C. W. and Wallis, W. D., Lecture Notes in Mathematics, 829, Springer-Verlag, 1980, 238–245).Google Scholar
[26]Vanstone, S. A. and Mullin, R. C., ‘A note on existence of weighing matrices W(22n−j, 2n) and associated combinatorial desings’, Utilitas Math. 8 (1975), 371–381.Google Scholar
[27]Wallis, J. Seberry, ‘Hadamard Matrices’ (Part IV of Combinatorics: Room sequences, sum free sets and Hadamard imatrices, by Wallis, W. D., Street, Anne Penfold and Wallis, Jennifer Seberry, in Lecture Notes in Mathematics 292, Springer-Verlag, 1972, 273–489).CrossRefGoogle Scholar
[28]Wilson, R. M., ‘Symmetric group divisible designs’, unpublished manuscript.Google Scholar
[29]Yates, F., ‘Complex experiments’, J. Royal Stat. Soc. B 2 (1935), 181–223.Google Scholar