Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-24T05:48:16.740Z Has data issue: false hasContentIssue false
  • English
  • Français

Hadamard matrices of order 36 with automorphisms of order 17

Published online by Cambridge University Press:  22 January 2016

Vladimir D. Tonchev
Vladimir D. Tonchev*
Affiliation:
Institute of Mathematics, Sofia 1090, P.O. Box 373, Bulgaria
Rights & Permissions [Opens in a new window]

Extract

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.

A Hadamard matrix of order n is an n by n matrix of 1’s and − 1’s such that HHt − nI. In such a matrix n is necessarily 1, 2 or a multiple of 4. Two Hadamard matrices H1 and H2 are called equivalent if there exist monomial matrices P, Q with PH1Q = H2. An automorphism of a Hadamard matrix H is an equivalence of the matrix to itself, i.e. a pair (P, Q) of monomial matrices such that PHQ = H. In other words, an automorphism of H is a permutation of its rows followed by multiplication of some rows by − 1, which leads to reordering of its columns and multiplication of some columns by − 1. The set of all automorphisms form a group under composition called the automorphism group (Aut H) of H. For a detailed study of the basic properties and applications of Hadamard matrices see, e.g. [1], [7, Chap. 14], [8].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 104 , December 1986 , pp. 163 - 174
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1986

References

[ 1 ] Th. Beth, D. Jungnickel, H. Lenz, Design theory, Bibliographisches Institut Mannheim/Wien/Zürich, 1985.Google Scholar
[ 2 ] Bussemaker, F. C. and Seidel, J. J., Symmetric Hadamard matrices of order 36, Ann. N.Y. Acad. Sci., 175 (1970), 66–79.Google Scholar
[ 3 ] Colbourn, M. J. and Mathon, R. A., On cyclic Steiner 2-designs, Ann. Discrete Math., 7 (1980), 215–254.Google Scholar
[ 4 ] Gibbons, P. B., Computing techniques for the construction and analysis of block designs, Technical Report, No. 92, Univ. of Toronto, 1976.Google Scholar
[ 5 ] Hall, M. Jr., Hadamard matrices of order 16, Jet Propulation Laboratory Research Summary No. 36–10, 1 (1961), 21–26.Google Scholar
[ 6 ] Hall, M. Jr., Hadamard matrices of order 20, Jet Propulation Technical Report No. 32–761, 1965.Google Scholar
[ 7 ] Hall, M. Jr., Combinatorial Theory, Gion (Blaisdell), Boston, 1967.Google Scholar
[ 8 ] Hedayat, A. and Wallis, W. D., Hadamard matrices and their applications, Ann. Statist., 6 (1978), 1184–1238.CrossRefGoogle Scholar
[ 9 ] Ito, N., Hadamard matrices with “doubly transitive” automorphism groups, Arch. Math., 35 (1980), 100–111.Google Scholar
[10] Ito, N., and Leon, J. S., An Hadamard matrix of order 36, J. Combin. Theory, A 34 (1983), 244–247.Google Scholar
[11] Ito, N., Leon, J. S. and Longyear, J. Q., Classification of 3-(24, 12, 5) designs and 24-dimensional Hadamard matrices, J. Combin. Theory, A 31 (1981), 66–93.CrossRefGoogle Scholar
[12] Kantor, W. M., Automorphism groups of Hadamard matrices, J. Combin. Theory, 6 (1969), 279–281.Google Scholar
[13] Norman, C. W., Non-isomorphic Hadamard designs, J. Combin. Theory, A 21 (1976), 366–344.CrossRefGoogle Scholar
[14] Paley, R. E. A. C., On orthogonal matrices, J. Math. Phys. MIT, 12 (1933), 311–320.Google Scholar
[15] Sims, C. C., Computational methods in the study of permutation groups, in “Computational Problems in Abstract Algebra” (Leech, J., Ed.), pp. 169–184, Pergamon, Elmsford, N.Y., 1970.Google Scholar
[16] Tonchev, V. D., Quasi-residual designs, codes and graphs, Colloq. Math. Soc. Janos Bolyai, 37 (1981), 685–695.Google Scholar
[17] Tonchev, V. D., Hadamard matrices of order 28 with automorphisms of order 13, J. Combin. Theory, A 35 (1983), 43–57.CrossRefGoogle Scholar
[18] Tonchev, V. D., Hadamard matrices of order 28 with automorphisms of order 7, J. Combin. Theory, Ser. A, 40 (1985), 62–81.Google Scholar
[19] Tonchev, V. D., Raev, R. V., Cyclic 2-(17, 8, 7) designs and related doubly-even codes, Compt. rend. Acad. bulg. Sci., 35 (1982), 1367–1370.Google Scholar
[20] Wallis, W. D., Hadamard equivalence, Congressus Numerantium, 28 (1980), 15–25.Google Scholar