A counterexample to Beder's conjectures about Hadamard matrices |
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Authors: | Dursun A Bulutoglu David M Kaziska |
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Institution: | Air Force Institute of Technology, WPAFB, OH 45433-7765, USA |
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Abstract: | In this note we provide a counterexample which resolves conjectures about Hadamard matrices made in this journal. Beder 1998. Conjectures about Hadamard matrices. Journal of Statistical Planning and Inference 72, 7–14] conjectured that if H is a maximal m×n row-Hadamard matrix then m is a multiple of 4; and that if n is a power of 2 then every row-Hadamard matrix can be extended to a Hadamard matrix. Using binary integer programming we obtain a maximal 13×32 row-Hadamard matrix, which disproves both conjectures. Additionally for n being a multiple of 4 up to 64, we tabulate values of m for which we have found a maximal row-Hadamard matrix. Based on the tabulated results we conjecture that a m×n row-Hadamard matrix with m?n-7 can be extended to a Hadamard matrix. |
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Keywords: | Hadamard matrix Maximal row-Hadamard matrix Binary integer programming |
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