CONSTRUCTION OF WEIGHING MATRICES AND HADAMARD MATRICES

CHANDRADEO PRASAD

Assistant Professor, Department. of CSE, RGIT, Koderma

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 . 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.

Abbreviations- H-matrices = Hadamard matrices, C-Matrix =

I. INTRODUCTION

Let us begin with the following definitions and basic facts: Dephased (normalized) H-matrix: A complex real 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 I. Thus if the matrix has an order n then CTC = (n-1) I.

If N> 1, there are two kinds of conference matrices. Let us normalize C by negating any row or column Which is not a conference matrix but this is quasi- whose first entry is negative. Thus a normalized symmetric. 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 symmetric (as is C if the first row is negated), or n is Complex weighing matrix: A matrix W = (n, k, C4) congruent to 2 (modulo 4) and S is symmetric (as is of order n with entries 0, ±1,± will be called C). 1 for everyi = 1, 2, 3... N. In a given dephased complex weighing matrix if WWT = kIn where WT matrix H, the lower right (N–1) × (N– 1) submatrix is stands for Hermitian conjugate of W and k is a called the core of H. positive number.

Weighing matrix: A weighing matrix W of order n Complimentary weighing matrices: Two real or with weight w is an n × n (0, 1, -1) matrix such that complex weighing matrices W1= [mij] and W2 = [nij] WWT = wI, where WT stands for of W. will be called complimentary if

A weighing matrix is called regular if its row and column sums are equal, and quasi-symmetric if its pattern zeros is symmetric.

A conference matrix of order n is a weighing matrix II. CONSTRUCTION ALGORITHMS W (n,n-1) but the converse is not true. For example consider the following symmetric weighing matrix W 1.2.1 CONTRUCTION OF A NEW WEIGHING (8,8) MATRIX FROM TWO WEIGHING MACHINES

Proceedings of 19th IRF International Conference, 25th January 2015, Chennai, India, ISBN: 978-93-84209-84-1 49 Construction of Weighing Matrices and Hadamard Matrices ALGORITHM 1 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 This proves the algorithm. FROM THE BUTSON 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

Proceedings of 19th IRF International Conference, 25th January 2015, Chennai, India, ISBN: 978-93-84209-84-1 50 Construction of Weighing Matrices and Hadamard Matrices 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.

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Proceedings of 19th IRF International Conference, 25th January 2015, Chennai, India, ISBN: 978-93-84209-84-1 51