Geodesic
Orthogonal Matrices with Zero Diagonal. II
Canadian journal of mathematics, Tome 23 (1971) no. 5, pp. 816-832

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

C-matrices appear in the literature at various places; for a survey, see [11]. Important for the construction of Hadamard matrices are the symmetric C-matrices, of order v ≡ 2 (mod 4), and the skew C-matrices, of order v ≡ 0 (mod 4). In § 2 of the present paper it is shown that there are essentially no other C-matrices. A more general class of matrices with zero diagonal is investigated, which contains the C-matrices and the matrices of (v, k, λ)-systems on k and k + 1 in the sense of Bridges and Ryser [6]. Skew C-matrices are interpreted in § 3 as the adjacency matrices of a special class of tournaments, which we call strong tournaments. They generalize the tournaments introduced by Szekeres [24] and by Reid and Brown [21]. 
Delsarte, P.; Goethals, J. M.; Seidel, J. J. Orthogonal Matrices with Zero Diagonal. II. Canadian journal of mathematics, Tome 23 (1971) no. 5, pp. 816-832. doi: 10.4153/CJM-1971-091-x
@article{10_4153_CJM_1971_091_x,
     author = {Delsarte, P. and Goethals, J. M. and Seidel, J. J.},
     title = {Orthogonal {Matrices} with {Zero} {Diagonal.} {II}},
     journal = {Canadian journal of mathematics},
     pages = {816--832},
     year = {1971},
     volume = {23},
     number = {5},
     doi = {10.4153/CJM-1971-091-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1971-091-x/}
}
TY  - JOUR
AU  - Delsarte, P.
AU  - Goethals, J. M.
AU  - Seidel, J. J.
TI  - Orthogonal Matrices with Zero Diagonal. II
JO  - Canadian journal of mathematics
PY  - 1971
SP  - 816
EP  - 832
VL  - 23
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1971-091-x/
DO  - 10.4153/CJM-1971-091-x
ID  - 10_4153_CJM_1971_091_x
ER  - 
%0 Journal Article
%A Delsarte, P.
%A Goethals, J. M.
%A Seidel, J. J.
%T Orthogonal Matrices with Zero Diagonal. II
%J Canadian journal of mathematics
%D 1971
%P 816-832
%V 23
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1971-091-x/
%R 10.4153/CJM-1971-091-x
%F 10_4153_CJM_1971_091_x

[1] 1. Assmus, E. F., Jr., and Mattson, H. F., Jr., On the automorphism groups of Paley-Hadamard matrices, Combinatorial mathematics and its applications, Bose, R. C. and Dowling, T. A. (Ed.), (University of North Carolina Press, Chapel Hill, 1969), 98–103. Google Scholar

[2] 2. Belevitch, V., Synthesis of four-wire conference networks and related problems, Proc. symp. on modern network synthesis, Polytechnic Institute of Brooklyn (1955), 175–195. Google Scholar

[3] 3. Belevitch, V., Conference networks and Hadamard matrices, Ann. Soc. Sci. BruxellesSér. 1 82 (1968), 13–32. Google Scholar

[4] 4. Berlekamp, E. R., Algebraic Coding Theory (McGraw-Hill, New York, 1968). Google Scholar

[5] 5. Berlekamp, E. R., Lint, J. H. van and Seidel, J. J., A strongly regular graph derived from the perfect ternary Golay code (to appear). Google Scholar

[6] 6. Bridges, W. G. and Ryser, H. J., Combinatorial designs and related systems, J. Algebra 13 (1969), 432–446. Google Scholar

[7] 7. Butson, A. T., Relations among generalized Hadamard matrices, relative difference sets, and maximal length linear recurring sequences, Can. J. Math. 15 (1963), 42–48. Google Scholar

[8] 8. Carmichael, R. D., Introduction to the theory of groups of finite order (Dover, London, 1956). Google Scholar

[9] 9. Dembowski, P., Finite Geometries (Springer-Verlag, New York, 1968). Google Scholar | DOI

[10] 10. Elliott, J. E. H. and Butson, A. T., Relative difference sets, Illinois J. Math. 10 (1966), 517–531. Google Scholar

[11] 11. Goethals, J. M. and Seidel, J. J., Orthogonal matrices with zero diagonal, Can. J. Math. 19 (1967), 1001–1010. Google Scholar

[12] 12. Goethals, J. M. and Seidel, J. J., A skew Hadamard matrix of order 36, J. Australian Math. Soc. 11 (1970), 343–344. Google Scholar

[13] 13. Hall, M., Jr., Cyclic projective planes, Duke Math. J. 14 (1947), 1079–1090. Google Scholar

[14] 14. Hall, M., Jr., Note on the Mathieu group M12, Arch. Math. 13 (1962), 334–340. Google Scholar

[15] 15. Hall, M., Jr., Combinatorial theory (Blaisdell, Waltham, Mass., 1967). Google Scholar

[16] 16. Kantor, W. M., Automorphism groups of Hadamard matrices, J. Combinatorial Theory 6 (1969), 279–281. Google Scholar

[17] 17. Lint, J. H. van and Seidel, J. J., Equilateral point sets in elliptic geometry, Nederl. Akad. Wetensch. Proc. Ser. A 69 (1966), 335–348. Google Scholar

[18] 18. Mann, H. B., Addition theorems (Wiley, New York, 1965). Google Scholar

[19] 19. Paley, R. E. A. C., On orthogonal matrices, J. Math, and Phys. 12 (1933), 311–320. Google Scholar

[20] 20. Raghavarao, D., Some aspects of weighing designs, Ann. Math. Statist. 31 (1960), 878–884. Google Scholar

[21] 21. Reid, K. B. and Brown, E., Doubly regular tournaments are equivalent to skew Hadamard matrices(to appear). Google Scholar

[22] 22. Seidel, J. J., Strongly regular graphs with (–1, 1, 0) adjacency matrix having eigenvalue 3, Linear Algebra and Appl. 1 (1968), 281–298. Google Scholar

[23] 23. Seidel, J. J., Strongly regular graphs, Recent progress in combinatorics, Tutte, W. T. (Ed.), Proc. Third Waterloo Conference on Combinatorics, 185–198 (Academic Press, New York, 1969). Google Scholar

[24] 24. Szekeres, G., Tournaments and Hadamard matrices, Enseignement.Math. 15 (1969), 269–278. Google Scholar

[25] 25. Wallis, J., Some (1, - 1 )>Matrices, J. Combinatorial Theory Ser. B 10 (1971), 1–11. Google Scholar

Cité par Sources :