AN ALTERNATIVE METHOD FOR CONSTRUCTING HADAMARD MATRICES

Symmetric Hadamard matrices are investigated in this research and an alternative method of construction is introduced. Using the proposed method, we can construct Hadamard matrices of order where and . This construction can be used to construct an infinite number of Hadamard matrices. For the present study, we use quadratic non-residues over a finite field.


INTRODUCTION
A Hadamard matrix of order whose rows and columns are mutually orthogonal with entries and satisfying , where is the transpose of and is the identity matrix of order n [1]. French mathematician Jacques Hadamard proved that such matrices could exist only if is 1,2 or a multiple of 4 [2]. Still there are unknown Hadamard matrices of order of multiple of 4. If , then is called symmetric Hadamard matrix. These matrices can be transformed to produce incomplete block design, t-design, error correcting and detecting codes, and other mathematical and statistical objects [3]. Hadamard  In 1893, Jacques Hadamard introduced Hadamard matrices of order and . He introduced his matrices when studying how large the determinant of a square matrix can be [5]. Another popular construction of Hadamard matrices were due to the English Mathematician Raymond Paley. He gave construction methods for various infinite classes of Hadamard matrices. The Paley construction is a method for constructing Hadamard matrices using finite fields [6]. This method uses quadratic residues in where is a power of an odd prime number, is a Galois field of order . An element in is a quadratic residue if and only if there exists in such that . Otherwise, is quadratic non-residue. Paley define quadratic character indicates whether the given finite field element is a perfect square or not. Hadamard matrix of size (Here, we denote -1 bysign). Where is a symmetric matrix of order q ( constructed using ) and is a column vector of length with all entries 1. Also, symmetric matrix has the properties and , where, is the matrix with all entries 1.
Another popular construction was discovered by John Williamson in 1944 which are generalizations of some of Paley's work. He constructed Hadamard matrices of order using four symmetric circulant matrices of order with entries and satisfying both, , for and [7].
In 1970, Symmetric Hadamard matrices of order were constructed by [8] Bussemaker and Seidel and Symmetric conference matrices of order 46 were constructed by R. Mathon in 1978 [9].
A conference matrix is a square matrix with on the diagonal and on the off diagonal such that is a multiple of the identity matrix . Thus, if the matrix has order , . There are some relations between conference matrices and Hadamard matrices of order . But not all conference matrices represent Hadamard matrices since conference matrices of size exist.
In 2014, by modifying Mathon's construction, Balonin and Seberry have constructed symmetric conference matrices of order 46 [10]. It is inequivalent to those Mathon. If two Hadamard matrices ( and with same order) are said to be equivalent, if can be obtained from by permuting rows and columns and by multiplying rows and columns by -1. Up to equivalence a unique Hadamard matrix of order 1, 2, 4, 8 and 12 exists [11]. Matteo, Dokovic and Kotsireas constructed symmetric Hadamard matrices of order [12]. All of them are constructed by using the GP array of Balonin and Seberry. Moreover, Kharaghani and Tayfeh discovered Hadamard matrix of order using T-sequences [13]. Now unknown smallest order Hadamard matrix is for skew-Hadamard matrices, and for symmetric Hadamard matrices [14].
In this paper we propose an alternative method of constructing symmetric Hadamard matrices using quadratic non-residues over finite fields.

MATERIAL AND METHODS
First, we define a function, as follows. It indicates whether the given finite field element a is a perfect square or not.
Let be the matrix whose rows and columns are indexed by elements of and construct using .
The matrix is Symmetric matrix of order with zero diagonal and elsewhere. Also, symmetric matrix has the properties and Where, is the matrix with all entries 1.

Method:
Let . We can get, and Therefore, is a symmetric Hadamard matrix of order .

Example III
Now consider (quadratic non-residues are 2,5,6,7,8 and 11) and . We can get, and Therefore, is a symmetric Hadamard matrix of order .

RESULTS AND DISCUSSION
Using the proposed method, we can construct symmetric Hadamard matrix of order where and .

CONCLUSIONS
The proposed alternative method which is our main result, can be used to construct an infinite number of Hadamard matrices. In this work, we used quadratic non-residues over a finite field. Using proposed method, we can construct symmetric Hadamard matrix of order where and . As a future work, planning to implement a computer programme to prove our method and construct large symmetric Hadamard matrices of order .