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Hadamard Matrices, Orthogonal Designs and Construction Algorithms
"... We discuss algorithms for the construction of Hadamard matrices. We include discussion of construction using Williamson matrices, Legendre pairs and the discret Fourier transform and the two circulants construction. Next we move to algorithms to determine the equivalence of Hadamard matrices using t ..."
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We discuss algorithms for the construction of Hadamard matrices. We include discussion of construction using Williamson matrices, Legendre pairs and the discret Fourier transform and the two circulants construction. Next we move to algorithms to determine the equivalence of Hadamard matrices using the pro le and projections of Hadamard matrices. A summary is then given which considers inequivalence of Hadamard matrices of orders up to 44. The nal two sections give algorithms for constructing orthogonal designs, short amicable and amicable sets for use in the Kharaghani array. 1 Algorithms for constructing Hadamard matrices 1.1 Hadamard matrices constructed from Williamson matrices An Hadamard matrix H of order n has elements 1 and satis es HH T = nI n . These matrices are used extensively in coding and communications (see Seberry and Yamada [90]). The order of an Hadamard matrix is 1, 2 or n (0 mod 4). The rst unsolved case is order 428. We use Williamson's construction as the basis of our algorithm to construct a distributed computer search for new Hadamard matrices. We briey describe the theory of Williamson's construction below. Previous computer searches for Hadamard matrices using Williamson's condition 2 are described in Section 1.1.1. The implementation of the search algorithm is presented in Section 1.1.2, and the results of the search are described in Section 1.1.3. Theorem 1 (Williamson [104]) Suppose there exist four (1; 1) matrices A, B, C, D of order n which satisfy XY T = Y X T ; X;Y 2 fA; B; C; Dg Further, suppose AA T +BB T + CC T +DD T = 4nI n (1) Then H = 2 6 6 6 4 A B C D B A D C C D A B D C B A 3 7 7 7 5 (2) is an Hadamard matrix of order 4n constructed from a Williamson array. Let the matrix T given below be called ...
Nonexistence results for Hadamardlike 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 rst row has no ze ..."
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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 rst row has no zeros, then n is not of the form p , 2p or 3p where p is an odd prime, and k is a positive integer.
WilliamsonHadamard spreading sequences for DSCDMA applications
, 2003
"... this paper, we apply the same technique to improve crosscorrelation properties of WilliamsonHadamard sequences. As it is always the case, the improvement is achieved at the expense of slightly worsening the autocorrelation properties. However, the overall autocorrelation properties of the modifie ..."
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this paper, we apply the same technique to improve crosscorrelation properties of WilliamsonHadamard sequences. As it is always the case, the improvement is achieved at the expense of slightly worsening the autocorrelation properties. However, the overall autocorrelation properties of the modified sequence sets are still significantly better than those of WalshHadamard sequences of comparable lengths
SUPPLEMENTARY DIFFERENCE SETS WITH SYMMETRY FOR HADAMARD MATRICES
, 903
"... Abstract. An overview of the known supplementary difference sets (SDSs) (Ai), 1 ≤ i ≤ 4, with parameters (n; ki; λ), ki = Ai, where each Ai is either symmetric or skew and ∑ ki = n + λ is given. Five new Williamson matrices over the elementary abelian groups of order 5 2, 3 3 and 7 2 are construct ..."
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Abstract. An overview of the known supplementary difference sets (SDSs) (Ai), 1 ≤ i ≤ 4, with parameters (n; ki; λ), ki = Ai, where each Ai is either symmetric or skew and ∑ ki = n + λ is given. Five new Williamson matrices over the elementary abelian groups of order 5 2, 3 3 and 7 2 are constructed. New examples of skew Hadamard matrices of order 4n for n = 47, 61, 127 are presented. The last of these is obtained from a (127, 57, 76) difference family that we have constructed. An old nonpublished example of Gmatrices of order 37 is also included. 2000 Mathematics Subject Classification 05B20, 05B30 1.