Method of Construction and Some Properties of 4-Row-Regular Circulant Partial Hadamard Matrices of Order ( k × 2 k )

. In this paper, some properties of circulant partial Hadamard matrices of the form 4 − H ( k × 2 k ) have been obtained together with a method of construction with the help of Toeplitz matrices


Introduction
A square matrix H of order n is called Hadamard matrix if HH ′ = nI n .Hadamard matrices are row orthogonal matrices.The order of a Hadamard matrix is 1 or 2 or 4t where t is a positive integer.Whether there exists a Hadamard matrix of order n for a given positive integer n which is divisible by 4? This is a challenging question for the mathematical community and is an open problem in mathematics known as the Hadamard matrix conjecture.Despite a large number of methods for constructing Hadamard matrices, the Hadamard matrix conjecture remains one of the most long-standing problems [4,8].Row orthogonal rectangular matrices of order k × n are known as partial Hadamard matrices.Partial Hadamard matrices are a generalization of Hadamard matrices.The partial Hadamard matrices which are formed by the right cyclic shift of its row are known as circulant partial Hadamard matrices.The known circulant Hadamard matrices are of order 4. Ryser [12] conjectured that there does not exist any circulant partial Hadamard matrix of order greater than 4 (see also, Euler [2]).This led to concentrating the study on circulant partial Hadamard matrices.In 2013, Craigen et al. [1] have been obtained many new results on circulant partial Hadamard matrices including a table which gives the maximum number of rows possible in a circulant partial Hadamard matrices up to order 64.The table obtained by Craigen et al. [1] is by computer search.Kao [5,6]  The first method of construction of this type of circulant partial Hadamard matrices is given by Manjhi and Rana [10].This article also provides some new properties of circulant partial Hadamard matrices including a new method of construction with the help of Toeplitz matrices.

Preliminaries
A square matrix H of order n and with entries ±1, is called a Hadamard matrix if Some examples of Hadamard matrices are given below: The latest construction of the new Hadamard matrix of order 428 is given by Kharaghani and Tayfeh-Rezaie in 2004 ( [7]).Orders of Hadamard matrices less than 1000 that are still unknown are 668, 716, 892 and 956.
A matrix M of order k and with entries 0, ±1 is called a conference matrix if MM ′ = (k − 1)I k .
Following are the only known circulant matrices which are Hadamard matrices: The circulant Hadamard matrices obtained so far are of order 1 or 4. H.J. Ryser [12] conjectured that there is no circulant partial Hadamard matrix of order greater than 4.
A rectangular circulant matrix H of order k × n with row sum r is called circulant partial Hadamard matrix if HH ′ = nI k , and is denoted by r − H(k × n).
Craigen et al. [1] used the notation r − H(k × n) for the class of all r-row-regular circulant Hadamard matrices of order (k× n).In this paper, the symbol r − H(k× n) is used for a particular member of the family of all r-row-regular circulant Hadamard matrices of order (k×n).Following are some examples of circulant partial Hadamard matrices: The properties of circulant partial Hadamard matrix r − H(k × n) related to proposed research work are given below: (vi) r k ≤ n (Cauchy-Schwartz inequality).
(vii) The sum of all column sums in H is equal to rk.
(viii) The sum of squares of column sums in H is equal to kn.
Particularly, for the circulant partial Hadamard matrices 2 − H(k × 2k), Craigen et al. [1] have given the following two important theorems and one conjecture: ) is equivalent to the existence of negacyclic conference matrix of order k.

Merging of Toeplitz Matrices to form Circulants
In this section some properties of Toeplitz matrices are discussed together with the sufficient conditions for merging two of more Toeplitz matrices to form a circulant matrix.

Consider a Toeplitz matrix T given by
.
Observe that From the above observation, we have the following theorem: Theorem 3. Every Toeplitz matrix can be expressed as a sum of a semicirculant matrix and a transpose of a semicirculant matrix.
We note that If T is any Toeplitz matrix of order n then T = αI n + S 1 + S ′ 2 , where S 1 and S 2 are semicirculant matrices with zero diagonal.Let us call this representation of T, the formal representation of T.
Next, theorem gives a sufficient condition that the matrix (A|B) is to be a circulant matrix for any two Toeplitz matrices A and B.
Theorem 5. Let A and B be two Toeplitz matrices each order n with the formal representations

Construction of Circulant Partial Hadamard Matrices 4 − H(k × 2k)
In 2013, R. Kragen et al. [1]  In this section a method of construction of circulant partial Hadamard matrices of the type 4 − H(k × 2k) is forwarded.This method merges four Toeplitz matrices of some special type to form a circular partial Hadamard matrix of the type 4 − H(k × 2k).The details are explained through the following theorem: Theorem 6.Let A, B, C and D be square Toeplitz matrices, each of order k 2 having diagonal entries zero and non-diagonal entries ±1.In addition, the matrices A, B, C and D satisfy the following properties: (ii) The augmented matrix (A|B|C|D) is a circulant.
Then there exists a circulant partial Hadamard matrix 4 − H(k × 2k) of the form Proof.(i) and (ii) implies that H is a circulant matrix with row sum 4. Next, where the block matrices X and Y are given by the following expressions: Using (i), (iii) and (iv), we get and We observe that the augmented matrix (A|B|C|D) is a circulant.Thus, the conditions of is a circulant partial Hadamard matrix of the type 4 − H(4 × 8).
Hence on the basis of the conditions given in Theorem 6, the matrix H given by .

More Results on Circulant Partial Hadamard Matrix 4 − H(k × 2k)
In this section some theorems and examples on circulant partial Hadamard matrix 4− H(k × 2k) are given with the help of column sum properties.

Thoughts and Conclusions
A general method of constructing a circulant partial Hadamard matrix of the form 4 − H(k × 2k) with the help of four Toeplitz matrices is proposed in Theorem 6.To date there is no method to construct circulant partial Hadamard matrices of the type 4 − H(k × 2k).Therefore, it will now become a routine task for researchers to search for Toeplitz matrices, from which we can construct some unknown circulant partial Hadamard matrices of the form 4 − H(k × 2k).
identified applications of circulant partial Hadamard matrices in the construction of functional design resonance imaging (fMRI) experimental designs.This article gives a method of construction of circulant partial Hadamard matrices of type 4 − H(k × 2k).The only known circulant partial Hadamard matrices of this type are for k = 6 and k = 14.Recently, some new examples of this type of circulant partial Hadamard matrices are given by Manjhi and Rana [9], but not of new unknown orders.

2 =Illustration 1 (
2kI k .This proves that H is a circulant partial Hadamard matrix of order (k × 2k) and with row sum 4. Construction of 4 − H(4 × 8)).Consider the following four zero-diagonal Toeplitz matrices each of order 2 with non-diagonal entries ±1:

Theorem 6
are satisfied by the matrices A, B, C and D.

Theorem 8 and
Theorem 9  give idea about the non-existence of column sums in a circulant partial Hadamard matrices of the type 4 − H(k × 2k).Section 2 includes some merging results on Toeplitz matrices.The results in Section 2 may give some information about the search for matrices A, B, C and D given in Theorem 6.The results obtained in this paper are obtained analytically and without a computer search.I hope, that the results will help the researchers in advancement of knowledge.

Question 1 .Question 2 .
On the basis of presented work herewith I am forwarding the following new questions: Does the existence of 4 − H(k × 2k) imply that k − 1 is an odd prime number?Does the divisibility of k by 4 indicate the non-existence of the partial circulant Hadamard matrices 4 − H(k × 2k)?
forward a table for maximum values of k in a circulant partial Hadamard matrix r − H(k × 2k).With the help of this table, they got many interesting results on circulant