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Construction of Weighing Matrices and Hadamard Matrices

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Construction of Weighing Matrices and Hadamard Matrices International Journal of Advances in Electronics and Computer Science, ISSN: 2393-2835 Volume-2, Issue-3, March-2015 CONSTRUCTION OF WEIGHING MATRICES AND HADAMARD MATRICES CHANDRADEO PRASAD Assistant Professor, Department. of CSE, RGIT, Koderma E-mail: [email protected] Abstract- Recently weighing matrices have been found much beneficial to engineers working with satellite and digital communications. They have been found to have many similarities with perfect ternary arrays. These arrays have been frequently implemented in digital communications. Complex Hadamard matrices have applications in quantum information theory and quantum tomography. The purpose of this paper is to forward simple constructions for some of these matrices so that they can be used by engineers. This paper introduces a new generalization of matrix orthogonality. It has been shown that several classical as well as Hadamard matrices with circulant blocks can be obtained from generalized orthogonal matrices. The order of new complex H-matrices are 26,36, 50 and 82. Butson H-matrices are constructed from generalized orthogonal matrices. Keywords- H-matrices = Hadamard matrices, C-Matrix = Conference matrix I. INTRODUCTION Let us begin with the following definitions and basic facts: Dephased (normalized) H-matrix: A complex real Hadamard matrix H = [Hij] of order n is called dephased or normalized if H1i = Hi1= 1. Conference Matrix:A Conference matrix is a square matrix C with 0 on diagonal and +1 or -1 off diagonal, such that CTC is an integral multiple of the identity matrix I. Thus if the matrix has an order n T then C C = (n-1) I. If N> 1, there are two kinds of conference matrices. Which is not a conference matrix but this is quasi- Let us normalize C by negating any row or column symmetric. whose first entry is negative. Thus a normalized conference matrix has all 1’s in its first row and column, except for a zero in the top left corner. Let S be the matrix that remains when the first row and column of C are removed. Then either n is a multiple 4, and S is skew Complex weighing matrix: A matrix W = (n, k, C4) symmetric (as is C if the first row is negated), or n is of order n with entries 0, ±1,± will be called congruent to 2 (modulo 4) and S is symmetric (as is complex weighing matrix if WWT = kIn where WT C). 1 for everyi = 1, 2, 3... N. In a given dephased stands for Hermitian conjugate of W and k is a matrix H, the lower right (N–1) × (N– 1) submatrix is positive number. called the core of H. Complimentary weighing matrices: Two real or Weighing matrix: A weighing matrix W of order n complex weighing matrices W1= [mij] and W2 = [nij] with weight w is an n × n (0, 1, -1) matrix such that will be called complimentary if WWT = wI, where WT stands for transpose of W. A weighing matrix is called regular if its row and column sums are equal, and quasi-symmetric if its pattern zeros is symmetric. II. CONSTRUCTION ALGORITHMS A conference matrix of order n is a weighing matrix W (n,n-1) but the converse is not true. For example 1.2.1 CONTRUCTION OF A NEW WEIGHING consider the following symmetric weighing matrix W MATRIX FROM TWO WEIGHING MACHINES (8,8) ALGORITHM 1 Construction of Weighing Matrices and Hadamard Matrices 12 International Journal of Advances in Electronics and Computer Science, ISSN: 2393-2835 Volume-2, Issue-3, March-2015 Input: Let W1 and W2 be complementary weighing matrices of order n and weights k1 and k2 respectively. Output: is a weighing matrix W(2n, 2(k1+k2)). Step 1 : We have Step 5: From steps 3 and 4, it has been proved that ߮ is an algebra isomorphism. It can be easily shown that in addition to addition, usual product and multiplication by scalars, this isomorphism also takes into account of unary operation of conjugation. Conjugate of a + ib corresponds to transpose of the matrix under the isomorphism. Therefore, W = W (2n, 2 (k1 + k2)). Corollary: If k1 + k2= n, then W (2n, 2n) is a Hadamard matrix of order 2n. III. CONSTRUCTION OF WEIGHING MATRIX FROM THE BUTSON This proves the algorithm. HADAMARD MATRIX B(4, 2N) ALGORITHM 2 Input: Given a Butson H-matrix B (4, 2n) Output: W is a weighing matrix of weight 2n such that WWT = 2nI4n. IV. COMPLEX H-MATRIX FROM COMPLEX WEIGHING MATRIX ALGORITHM 3 Input: Given two complementary working matrices W1 and W2 with entries 0, ±1,± of order n and Construction of Weighing Matrices and Hadamard Matrices 13 International Journal of Advances in Electronics and Computer Science, ISSN: 2393-2835 Volume-2, Issue-3, March-2015 T weights k1, k2 such that k1 + k2 = n and W stands for [5] Craigen R and W D Wallis, Hadamard Matrices, 1893- Hermitian conjugate of W. 1993, Congr. Numer. 97(1993), 99-129. [6] Golomb S W and G Gong, Signal design for good correlation (2005), Cambridge university press, New York. of order 2n. [7] Hedayat A S, N J S Sloane and J Stufken, Orthogonal arrays, Theory and Applications, Springer, New York, 1999. Steps are analogous to that of ALGORITHM 1. [8] Koukouvinos C and Stalianou S “On Skew Hadamard Matrices”, Descrete Mathematics, V 308, 13, 2008, 2723- CONCLUSION 31. This paper shows that a weighing matrix can be [9] Mare Gysin and Jennifer Seberry, Construction Methods for weighing matrices of order 4n and weight 4n-2, 2n-1 and n obtained from any quaternary complex Hadamard using elementary properties of cyclotomy, In Conference on matrix as well as from two suitable disjoint weighing Combintorial Mathematics and Combinatorian Computing, matrices. Also it has been shown that Hadamard and University of Technology, Aukland, July,1994. complex Hadamard matrices can be obtained from [10] Mare Gysinand Jennifer Seberry, New Weighing Matrices certain complementary weighing matrices and through Linear Combinations of generalized cosets, In complex weighing matrices respectively. Conference on Combintorial Mathematics and Combinatorian Computing, University of Technology, Sydney, July,1996. REFERENCES [11] Craigen R, The Structure of weighing matrices having large [1] Arasu K.T., Leung, K.H., Ma S.L., Nabav A and weights, Spriger, Netherlands, Vol. 5, Number 3, May, Choudhary D. K. R. (2006) “Circulanant Weighing Matices 1995. of weight 22t, Journal of Designs, Codes and Cryptography, [12] Berman G, Families of generalized weighing matrices, Springer Neitherlands, Vol. 41, Number 1/October, 2006, Canadian J. Math., 30(1978), pp. 1016-1028. pp – 111-123. [13] Seberry Jennifer, Yamada M, “On product of Hadamard [2] Arasu K.T., W. de Launey and S. L. Ma “On circulant Matrices, Williamson Matrices and other orthogonal complex Hadamard matrices, Designs, Codes and matrices using M- Structures”, JCMCC 7?(1990), pp. 97- Cryptography,. 25(2002),123-142. 137. [3] Butson A T, Generalized Hadamard Matrices, Proc. Amer. [14] Seberry Jennifer, Yamada M (1992), “Hadamard Matrices, Math.Soc., 13(1962), 894-898. sequence and block designs, John Wiley and Sons Inc. pp. [4] Craigen R and HKharaghani, Hadamard Matrices from 431-437. weighing matrices vis signed groups, Des. Codes, [15] Seberry Jennifer, Website: http//www.uow.edu.au/ jennie/. Cryptography, 12(1997), 49-58. Construction of Weighing Matrices and Hadamard Matrices 14 .
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