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Báo cáo khoa học:Nonexistence results for Hadamard-like matrices

The class of square (0, 1,−1)-matrices whose rows are nonzero and mutually orthogonal is studied. This class generalizes the classes of Hadamard and Weighing matrices. We prove that if there exists an n by n (0, 1,−1)-matrix whose rows are nonzero, mutually orthogonal and whose first row has no zeros, then n is not of the form pk, 2pk or 3p where p is an odd prime, and k is a positive integer. | Nonexistence results for Hadamard-like matrices Justin D. Christian and Bryan L. Shader Department of Mathematics University of Wyoming USA christianjd@yahoo.com bshader@uwyo.edu Submitted Aug 26 2003 Accepted Jan 19 2004 Published Jan 23 2004 MR Subject Classifications 05B20 15A36 Abstract The class of square 0 1 1 -matrices whose rows are nonzero and mutually orthogonal is studied. This class generalizes the classes of Hadamard and Weighing matrices. We prove that if there exists an n by n 0 1 1 -matrix whose rows are nonzero mutually orthogonal and whose first row has no zeros then n is not of the form pk 2pk or 3p where p is an odd prime and k is a positive integer. 1 Introduction A Hadamard matrix of order n is an n by n 1 1 -matrix H satisfying HHT nl where I denotes the identity matrix and HT denotes the transpose of H. Hadamard matrices were first introduced by J. Hadamard in 1893 as solutions to a problem about determinants see GS WSW . The following well-known simple result shows that the standard necessary condition that is n 1 n 2 or n 0 mod 4 for the existence of a Hadamard matrix of order n is a consequence of the mutual orthogonality of three 1 1 -vectors. Proposition 1 Let u v and w be mutually orthogonal 1 by n 1 1 -vectors. Then n 0 mod 4. Proof. Each entry in the vectors u v and u w is even. Hence u v u w is a multiple of 4. Since u v u w u u n the result follows. The famous Hadamard Conjecture asserts that there exists a Hadamard matrix of order n for every n 0 mod 4 and has been verified for n 428 see HKS . Weighing matrices are generalizations of Hadamard matrices. Let n and w be positive integers. An n w -weighing matrix is an n by n 0 1 1 -matrix W wij satisfying WWT wl. Weighing matrices have been extensively studied see C and the references therein . Several necessary conditions for the existence of an n w -weighing matrix are known. If n 1 is odd then necessarily w is a perfect square and n w w 1 with THE ELECTRONIC JOURNAL OF .