Abstract
In this note we use combinatorial methods to show that the unique, up to equivalence, 5 ×5 (1, − 1)-matrix with determinant 48, the unique, up to equivalence, 6 ×6 (1, − 1)-matrix with determinant 160, and the unique, up to equivalence, 7 ×7 (1, − 1)-matrix with determinant 576, all cannot be embedded in the Hadamard matrix of order 8. We also review some properties of Sylvester Hadamard matrices, their Smith Normal Forms, and pivot patterns of Hadamard matrices when Gaussian Elimination with complete pivoting is applied on them. The pivot values which appear reconfirm the above non-embedding results.
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Bouyukliev, I., Fack, V., Winne, J.: 2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order, and their related Hadamard matrices and codes. Designs Codes Cryptogr. 51, 105 (2009)
Cryer, C.W.: Pivot size in Gaussian elimination. Numer. Math. 12, 335–345 (1968)
Day, J., Peterson, B.: Growth in Gaussian elimination. Amer. Math. Monthly 95, 489–513 (1988)
de Launey, W.: On the asymptotic existence of Hadamard matrices. http://arxiv.org/abs/1003.4001
Edelman, A., Mascarenhas, W.: On the complete pivoting conjecture for a Hadamard matrix of order 12. Linear Multilinear Algebra 38, 181–187 (1995)
Gould, N.: On growth in Gaussian elimination with pivoting. SIAM J. Matrix Anal. Appl. 12, 354–361 (1991)
Hall, M. Jr.: Hadamard matrices of order 16. J.P.L. Research Summary No.36-10, vol. 1, pp. 21–26 (1961)
Kharaghani, H., Orrick, W.: D-optimal matrices. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn., pp. 296–298. Chapman and Hall, Boca Racon (2007)
Kharaghani, H., Tayfeh-Rezaie, B.: On the Classification of Hadamard Matrices of Order 32. http://www3.interscience.wiley.com/journal/123249643/abstract
Kravvaritis, C., Mitrouli, M.: The growth factor of a Hadamard matrix of order 16 is 16. Numer. Linear Algebra Appl. 16(7–9), 715–743 (2009)
Macduffee, C.C.: The Theory of Matrices, 1st ed. Reprint. Chelsea, New York (1964)
Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Boston (1964)
Michael, T.S., Wallis, W.D.: Skew Hadamard matrices and the Smith normal form. Designs Codes Cryptogr. 13, 173–176 (1998)
Seberry, J., Xia, T., Koukouvinos, C., Mitrouli, M.: The maximal determinant and subdeterminants of ±1 matrices. Linear Algebra Appl. 373, 297–310 (2003)
Wallis, J.S.: Hadamard matrices, Part IV, of W.D. Wallis, Anne Penfold Street and Jennifer Seberry (Wallis). In: Combinatorics: Room Squares, Sum-Free Sets and Hadamard Matrices. LNM. Springer, Berlin (1972)
Smith, H.J.S.: Arithmetical notes. Proc. Lond. Math. Soc. 4, 236–253 (1873)
Spence, E.: A note on the equivalence of Hadamard matrices. Not. Am. Math. Soc. 18, 624 (1971)
Sylvester, J.J.: Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tesselated pavements in two or more colours, with applications to Newton’s rule, ornamental tile work, and the theory of numbers. Phil. Mag. 34(4), 461–475 (1867)
Wallis, W.D., Wallis, J.S.: Equivalence of Hadamard matrices. Israel J. Math. 7, 122–128 (1969)
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We thank the referees for their thoughtful and thought provoking comments, the references they gave, and the inspiration they provided.
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Dedicated with great respect to Warwick de Launey.
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Seberry, J., Mitrouli, M. Some remarks on Hadamard matrices. Cryptogr. Commun. 2, 293–306 (2010). https://doi.org/10.1007/s12095-010-0036-9
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DOI: https://doi.org/10.1007/s12095-010-0036-9
Keywords
- Hadamard matrices
- Smith normal form
- Embedding matrices
- Completely pivoted
- Determinant
- Gaussian elimination