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The Encyclopedia of Integer Sequences
"... This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs ..."
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This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other
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 ...
SWITCHING OPERATIONS FOR HADAMARD MATRICES
, 2005
"... Abstract. We define several operations that switch substructures of Hadamard matrices thereby producing new, generally inequivalent, Hadamard matrices. These operations have application to the enumeration and classification of Hadamard matrices. To illustrate their power, we use them to greatly impr ..."
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Abstract. We define several operations that switch substructures of Hadamard matrices thereby producing new, generally inequivalent, Hadamard matrices. These operations have application to the enumeration and classification of Hadamard matrices. To illustrate their power, we use them to greatly improve the lower bounds on the number of equivalence classes of Hadamard matrices in orders 32 and 36 to 3,578,006 and 4,745,357. 1.
On the enumeration of some Doptimal designs
 Journal of Statistical Planning and Inference
, 2008
"... Abstract. Two matrices with elements taken from the set {−1, 1} are Hadamard equivalent if one can be converted into the other by a sequence of permutations of rows and columns, and negations of rows and columns. In this paper we summarize what is known about the number of equivalence classes of mat ..."
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Abstract. Two matrices with elements taken from the set {−1, 1} are Hadamard equivalent if one can be converted into the other by a sequence of permutations of rows and columns, and negations of rows and columns. In this paper we summarize what is known about the number of equivalence classes of matrices having maximal determinant. We establish that there are 7 equivalence classes for matrices of order 21 and that there are at least 9,884 equivalence classes for matrices of order 26. The latter result is obtained primarily using a switching technique for producing new designs from old. 1.
On the classification of Hadamard matrices of order 32
, 2009
"... All equivalence classes of Hadamard matrices of order at most 28 have been found by 1994. Order 32 is where a combinatorial explosion occurs on the number of inequivalent Hadamard matrices. We find all equivalence classes of Hadamard matrices of order 32 which are of certain types. It turns out that ..."
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All equivalence classes of Hadamard matrices of order at most 28 have been found by 1994. Order 32 is where a combinatorial explosion occurs on the number of inequivalent Hadamard matrices. We find all equivalence classes of Hadamard matrices of order 32 which are of certain types. It turns out that there are exactly 13,680,757 Hadamard matrices of one type and 26,369 such matrices of another type. Based on experience with the classification of Hadamard matrices of smaller order, it is expected that the number of the remaining two types of these matrices, relative to the total number of Hadamard matrices of order 32, to be insignificant.
Extremal doubly even (56,28,12) codes and Hadamard matrices of order 28, Australas
 J. Combin
, 1994
"... Abstract. In [2] Bussemaker and Tonchev constructed six doubly even (56,28, 12) codes from two Hadamard matrices of order 28. But two of them were not distinguished. In [11] and [12] we characterized Hadamard matrices of order 28 and there are exactly 487 Hadamard matrices, up to equivalence. In thi ..."
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Abstract. In [2] Bussemaker and Tonchev constructed six doubly even (56,28, 12) codes from two Hadamard matrices of order 28. But two of them were not distinguished. In [11] and [12] we characterized Hadamard matrices of order 28 and there are exactly 487 Hadamard matrices, up to equivalence. In this paper we show that only two of the above 487 matrices produce six doubly even (56,28,12) codes and that two of the six codes are equivalent. Therefore there are exactly five (56,28,12) codes, up to equivalence, produced by Hadamard matrices of order 28. 1.
A SENSITIVE ALGORITHM FOR DETECTING THE INEQUIVALENCE OF HADAMARD MATRICES
"... Abstract. A Hadamard matrix of side n is an n × n matrix with every entry either 1 or −1, which satisfies HH T = nI. Two Hadamard matrices are called equivalent if one can be obtained from the other by some sequence of row and column permutations and negations. To identify the equivalence of two Had ..."
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Abstract. A Hadamard matrix of side n is an n × n matrix with every entry either 1 or −1, which satisfies HH T = nI. Two Hadamard matrices are called equivalent if one can be obtained from the other by some sequence of row and column permutations and negations. To identify the equivalence of two Hadamard matrices by a complete search is known to be an NP hard problem when n increases. In this paper, a new algorithm for detecting inequivalence of two Hadamard matrices is proposed, which is more sensitive than those known in the literature and which has a close relation with several measures of uniformity. As an application, we apply the new algorithm to verify the inequivalence of the known 60 inequivalent Hadamard matrices of order 24; furthermore, we show that there are at least 382 pairwise inequivalent Hadamard matrices of order 36. The latter is a new discovery. 1.
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"... Hadamard matrices, orthogonal designs and construction algorithms ..."
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The Lattice of NRun Orthogonal Arrays
, 2000
"... If the number of runs in a (mixedlevel) orthogonal array of strength 2 is specified, what numbers of levels and factors are possible? The collection of possible sets of parameters for orthogonal arrays with N runs has a natural lattice structure, induced by the "expansive replacement" con ..."
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If the number of runs in a (mixedlevel) orthogonal array of strength 2 is specified, what numbers of levels and factors are possible? The collection of possible sets of parameters for orthogonal arrays with N runs has a natural lattice structure, induced by the "expansive replacement" construction method. In particular the dual atoms in this lattice are the most important parameter sets, since any other parameter set for an Nrun orthogonal array can be constructed from them. To get a sense for the number of dual atoms, and to begin to understand the lattice as a function of N , we investigate the height and the size of the lattice. It is shown that the height is at most 1)#, where c = 1.4039 . . ., and that there is an infinite sequence of values of N for which this bound is attained. On the other hand, the number of nodes in the lattice is bounded above by a superpolynomial function of N (and superpolynomial growth does occur for certain sequences of values of N ). Using a new construction based on "mixed spreads", all parameter sets with 64 runs are determined. Four of these 64run orthogonal arrays appear to be new.
On Circulant Best Matrices and Their Applications
"... Call four type 1 (1; \Gamma1) matrices, X 1 ; X 2 ; X 3 ; X 4 , of the same group of order m (odd) with the properties (i) (X i \GammaI ) T = \Gamma(X i \GammaI ); i = 1; 2; 3 ; (ii) X T 4 = X 4 and the diagonal elements are positive, (iii) X i X j = X j X i and (iv) X 1 X T 1 +X 2 X T 2 +X 3 X T 3 ..."
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Call four type 1 (1; \Gamma1) matrices, X 1 ; X 2 ; X 3 ; X 4 , of the same group of order m (odd) with the properties (i) (X i \GammaI ) T = \Gamma(X i \GammaI ); i = 1; 2; 3 ; (ii) X T 4 = X 4 and the diagonal elements are positive, (iii) X i X j = X j X i and (iv) X 1 X T 1 +X 2 X T 2 +X 3 X T 3 +X 4 X T 4 = 4mIm ; best matrices. We use a computer to give, for the first time, all inequivalent best matrices of odd order m 31. Inequivalent best matrices of order m, m odd, can be used to find inequivalent skewHadamard matrices of order 4m. We use best matrices of order 1 4 (s 2 +3) to construct new orthogonal designs, including new OD(2s 2 +6; 1; 1; 2; 2; s 2 ; s 2 ). AMS Subject Classification: Primary 05B20, Secondary 05B30 Key words and phrases: Circulant matrices, supplementary difference sets, orthogonal designs, Hadamard matrices. 1