Geodesic
Partial λ-Geometries and Generalized Hadamard Matrices Over Groups
Canadian journal of mathematics, Tome 31 (1979) no. 3, pp. 617-627

Voir la notice de l'article provenant de la source Cambridge University Press

Section 1 of this paper contains all the work which deals exclusively with generalizations of Hadamard matrices. The non-existence theorem proven here (Theorem 1.10) generalizes a theorem of Hall and Paige [15] on the non-existence of complete mappings in certain groups.In Sections 2 and 3, we consider the duals of (Hanani) transversal designs; these dual structures, which we call (s, r, μ)-nets, are a natural generalization of the much studied (Bruck) nets which in turn are equivalent to sets of mutually orthogonal Latin squares. An (s, r, μ)-net is a set of s 2 μ points together with r parallel classes of blocks. Each class consists of s blocks of equal cardinality. Two non-parallel blocks meet in precisely μ points. It has been proven that r is always less than or equal to (s 2 μ – l) / (s – 1). 
Drake, David A. Partial λ-Geometries and Generalized Hadamard Matrices Over Groups. Canadian journal of mathematics, Tome 31 (1979) no. 3, pp. 617-627. doi: 10.4153/CJM-1979-062-1
@article{10_4153_CJM_1979_062_1,
     author = {Drake, David A.},
     title = {Partial {\ensuremath{\lambda}-Geometries} and {Generalized} {Hadamard} {Matrices} {Over} {Groups}},
     journal = {Canadian journal of mathematics},
     pages = {617--627},
     year = {1979},
     volume = {31},
     number = {3},
     doi = {10.4153/CJM-1979-062-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1979-062-1/}
}
TY  - JOUR
AU  - Drake, David A.
TI  - Partial λ-Geometries and Generalized Hadamard Matrices Over Groups
JO  - Canadian journal of mathematics
PY  - 1979
SP  - 617
EP  - 627
VL  - 31
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1979-062-1/
DO  - 10.4153/CJM-1979-062-1
ID  - 10_4153_CJM_1979_062_1
ER  - 
%0 Journal Article
%A Drake, David A.
%T Partial λ-Geometries and Generalized Hadamard Matrices Over Groups
%J Canadian journal of mathematics
%D 1979
%P 617-627
%V 31
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1979-062-1/
%R 10.4153/CJM-1979-062-1
%F 10_4153_CJM_1979_062_1

[1] 1. Bose, R. C., Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math. 13 (1963), 389–410. Google Scholar

[2] 2. Bose, R. C. and Bush, K. A., Orthogonal arrays of strength two and three, Annals Math. Stat. 23 (1952), 508–524. Google Scholar

[3] 3. Bose, R. C. and Connor, W. S., Combinatorial properties of group divisible incomplete block designs, Annals Math. Stat. 23 (1952), 367–383. Google Scholar

[4] 4. Bose, R. C., Shrikhande, S. S. and Singhi, N. M., Edge regular multigraphs and partial geometric designs with an application to the embedding of quasi-residual designs, Colloquio Internazionale sulle Teorie Combinatorie, Rome, 1973, atti dei convegni Lince. 17 (1976), 49–81. Google Scholar

[5] 5. Bruck, R. H., Finite nets I. numerical invariants, Can. J. Math. 3 (1951), 94–107. Google Scholar

[6] 6. Butson, A. T., Generalized Hadamard matrices, Proc. Amer. Math. Soc. 13 (1962), 894–898. Google Scholar

[7] 7. Connor, W. S., Some relations among the blocks of symmetrical group divisible designs, Annals Math. Stat. 23 (1952), 602–609. Google Scholar

[8] 8. Cameron, P. J., Parallelisms of complete designs (London Math. Soc. Lect. Note Series 23, Cambridge Univ. Press, 1976). Google Scholar

[9] 9. Cameron, P. J. and Drake, D. A., Partial X-geometries of small nexus, proceedings of the Symposium on Combinatorial Math, and Optimal Design, Fort Collins, 1978, to appear. Google Scholar

[10] 10. Dembowski, P., Finite geometries (Ergebnisse der Math, und ihrer Grenzgebiete 44, Springer- Verlag, Berlin-Heidelberg-New York, 1968). Google Scholar

[11] 11. Denes, J. and Keedwell, A. D., Latin squares and their applications (English Univ. Press Ltd., London, 1974). Google Scholar

[12] 12. Drake, D. A., Maximal sets of Latin squares and partial transversals, J. Stat. Planning and Inferenc. 1 (1977), 143–149. Google Scholar

[13] 13. Drake, D. A. and Jungnickel, D., Klingenberg structures and partial designs I: congruence relations and solutions, J. Stat. Planning and Inferenc. 1 (1977), 265–287. Google Scholar

[14] 14. Drake, D. A. and Jungnickel, D., Klingenberg structures and partial designs II: regularity and uniformity, Pacific J. Math. 77 (1978), 389–415. Google Scholar

[15] 15. Hall, M. and Paige, L. J., Complete mappings of finite groups, Pacific J. Math. 5 (1955), 541–549. Google Scholar

[16] 16. Hanani, H., On transversal designs, proceedings of the Advanced Study Institute on Combinatorics, Breukelen, 1974, Math. Centre Tracts 55, Amsterdam (1974), 42–52. Google Scholar

[17] 17. Jungnickel, D., On difference matrices, resolvable TD's and generalized Hadamard matrices, Math Zeit., to appear. Google Scholar

[18] 18. Plackett, R. L. and Burman, J. P., The design of optimum multifactorial experiments, Biometrik. 33 (1945), 305–325. Google Scholar

[19] 19. Rajkundlia, D., Some techniques for constructing new infinite families of balanced incomplete block designs, Math. Preprint No. 1978-16, Queen's University at Kingston, Ont. Google Scholar

[20] 20. Seberry, J., Some remarks on generalized Hadamard matrices and theorems of Rajkundlia on SBIBD's, Combinatorics VI : Proceedings of the SixthAustralian Conference, Armidale, 1978, to appear. Google Scholar

[21] 21. Wallis, W. D., Street, A. P. and Wallis, J. S., Combinatorics: Room squares, sum-free sets, Hadamard matrices, Lecture Notes in Math. 292 (Springer-Verlag, Berlin-Heidelberg-New York, 1972), 273–489. Google Scholar

[22] 22. Zassenhaus, H., The theory of groups (Chelsea Publ. Co., New York, New York, second edition, 1958). Google Scholar

Cité par Sources :