Some Nonexistence and Asymptotic Existence Results for Weighing Matrices

Abstract

Orthogonal designs and weighing matrices have many applications in areas such as coding theory, cryptography, wireless networking, and communication. In this paper, we first show that if positive integer cannot be written as the sum of three integer squares, then there does not exist any skew-symmetric weighing matrix of order and weight , where is an odd positive integer. Then we show that, for any square , there is an integer such that, for each , there is a symmetric weighing matrix of order and weight . Moreover, we improve some of the asymptotic existence results for weighing matrices obtained by Eades, Geramita, and Seberry.

1. Introduction

An orthogonal design (OD) [1, Chapter 1] of order and type , denoted as , is a square matrix of order with entries from , where ’s are commuting variables, that satisfies where is the transpose of and is the identity matrix of order . An OD with no zero entry is called a full OD. Equating all variables to in any full OD results in a Hadamard matrix. Equating all variables to in any OD of order results in a weighing matrix, denoted as , where is the weight that is the number of nonzero entries in each row (column) of the weighing matrix.

The Kronecker product of two matrices and of orders and , respectively, is denoted by , and it is the matrix of order defined by The direct sum of and is denoted by , and it is the matrix of order which is defined as follows: where represents a zero matrix of appropriate dimension.

Let and , where ’s and ’s belong to a commutative ring (e.g., ). Square matrix of order is called circulant if , where is reduced modulo . Square matrix of order is called back-circulant if , where is reduced modulo . We have and ; that is, any back-circulant matrix is symmetric. Let be a square matrix of order , where if modulo and otherwise. Matrix is called back-diagonal matrix. It is not hard to see that matrix is back-circulant and so symmetric (cf. [1, Chapter 4]).

A rational family of order and type , where ’s are positive rational numbers, is a collection of rational matrices of order , , that satisfy () and , (). is a skew-symmetric matrix if . Two matrices of the same dimension are disjoint if their entrywise multiplication is a zero matrix [1, Chapters 1, 2].

Geramita and Seberry [1, Chapter 7] showed some existence results for weighing matrices. They showed that when the orders of ODs and weighing matrices are much larger than the number of nonzero entries in each row, the necessary conditions for existence of ODs and weighing matrices are also sufficient. In this paper, we show some nonexistence results on weighing matrices and some asymptotic results for existence of weighing matrices.

2. Nonexistence Results for Weighing Matrices

In this section, we show some nonexistence results for weighing matrices. The results are summarized in Theorems 4 and 5 (known) and Theorem 10.

Lemma 1 (e.g., [2, Chapter 2]). The eigenvalues of a symmetric matrix with real entries are real.

Lemma 2 (e.g., [2, Chapter 2]). The eigenvalues of a skew-symmetric matrix with real entries are of the form , where b is a real number.

Lemma 3 (e.g., [1]). The absolute values of the eigenvalues of a weighing matrix are

Theorem 4 (e.g., [3]). There does not exist any symmetric weighing matrix with zero diagonal of odd order.

Proof. Suppose that odd, is a symmetric weighing matrix with zero diagonal. From Linear Algebra, , where ’s are eigenvalues of By Lemmas 1 and 3, since ,Since is odd, must be odd and therefore nonzero, but, by assumption, , which is a contradiction.

Theorem 5 (e.g., [3]). There is no skew-symmetric weighing matrix of odd order.

Proof. Assume that is a skew-symmetric weighing matrix of odd order. From Lemmas 2 and 3, eigenvalues of are in form Therefore, Since is odd, must be odd and so nonzero, but since is a skew-symmetric matrix, which is a contradiction.

Next, we show that if is any odd number and cannot be written as the sum of three integer squares, then there is no skew-symmetric weighing matrix . To do so, we first bring the following well known results.

Lemma 6 (e.g., [4]). A positive integer can be written as the sum of three integer squares if and only if it is not of the form , where

The following lemma is a useful result that can be concluded from Lemma 6, and, for the sake of completion, we bring its proof.

Lemma 7 (e.g., [4]). A positive integer is the sum of three rational squares if and only if it is the sum of three integer squares.

Proof. Suppose that a positive integer is the sum of three rational squares. Reducing the three rational numbers to the same denominator, one may write where , and are integers. Suppose that cannot be written as the sum of three integer squares. From Lemma 6, there exist nonnegative integers such that One may write as , for some nonnegative integers Thus, , where is a nonnegative integer, and so where . This is a contradiction because, by Lemma 6, cannot be written as the sum of three integer squares, whereas by assumption Therefore, the result follows.

Lemma 8 (Shapiro [5]). There is a rational family in order , odd, of type if and only if there is a rational family of the same type in order

Lemma 9 (Geramita and Seberry [1]). A necessary and sufficient condition that there is a rational family of type in order is that is a sum of three rational squares.

Proof. Suppose that is a rational family of type in order Then is also a rational family of the same type and order. Thus and Since is a skew-symmetric matrix, the diagonal of is zero, so is a sum of three rational squares.
Now let , where , and are rational numbers. If we let then is a rational family of type and order

We use Lemmas 7, 8, and 9 to prove the following nonexistence result.

Theorem 10. Suppose that positive integer cannot be written as the sum of three integer squares. Then there does not exist skew-symmetric , for any odd number

Proof. If there is skew-symmetric for some odd number , then is a rational family of type and order . Thus, by Lemma 8, there is a rational family of type and order 4. Lemmas 7 and 9 imply that must be the sum of three integer squares.

3. Asymptotic Existence Results of Weighing Matrices

In this section, we provide some asymptotic results for existence of weighing matrices. These results are summarized in Theorem 21 and Theorems 22 and 25 (for ODs) that use different methodologies to improve the known results shown by Geramita and Seberry [1, Chapter 7].

Lemma 11 (e.g., [1]). A necessary and sufficient condition that there exists is that there exists a family of pairwise disjoint square matrices of order with entries from satisfying(i) is a ,(ii)

The following lemma, due to Sylvester, is known, and we bring its proof.

Lemma 12 (see [6]). Let and be two relatively prime positive integers. Then every integer can be written in the form , where and are nonnegative integers.

Proof. Let be an integer greater than or equal to Since and are relatively prime, there are integers and such that (see [4]). So, where One can choose such that For such , we let and The condition implies that must be positive.

The following lemma shows how to construct ODs of higher orders by using two ODs of the same types but different orders. The first part of the lemma is known (cf. [1, Lemma 7.22]).

Lemma 13. Suppose that there are and Let Then there is an integer such that, for each , there is Moreover, if and are symmetric, then there is an integer such that, for each , there is symmetric .

Proof. Let and Then and are relatively prime. Let , and let be a positive integer By Lemma 12, there are nonnegative integers and such that Since there exist and , there are families of order and of order satisfying the conditions in Lemma 11. We define the family of order It can be seen that this family satisfies the conditions of Lemma 11; therefore it makes
Now if and are symmetric, then ’s and ’s, , are all symmetric. Since set consists of symmetric matrices of order satisfying the conditions of Lemma 11, and so they generate symmetric

Theorem 14 (Wallis and Whiteman [7]). Let be a prime power. Then there is circulant

Corollary 15 (see [1, 8]). Suppose that is a prime power and is any positive integer. Then there is circulant

Proof. Let be a fixed positive integer. From Theorem 14, we know that there exists a circulant Suppose that the first row of this matrix is Let Thus, , where is a primitive root of unity and For define where is the smallest integer greater than or equal to We show that if , then is To see this, let Thus we have Since , for all such that , for all such that Applying the finite Parseval relation where is reduced modulo , for gives And, for , . Therefore, is circulant

The next lemma shows how to make a symmetric OD to be used for Theorem 21.

Lemma 16. Let be a positive integer. Then there exists symmetric .

Proof. Define and, for , , where It can be directly verified that the family of order satisfies the conditions of Lemma 11, and therefore it makes symmetric . Note that , and are symmetric.

Theorem 17 (Robinson [9]). All exist, where and .

We prove the following well known lemma by giving a proof which is different from the proof in [1, Lemma 7.27].

Lemma 18. For any sequence of positive integers, there is a positive integer such that there is skew-symmetric .

Proof. Let and be the smallest positive integers such that and . By Theorem 17, there are and Without loss of generality, assume that and are two families corresponding to and satisfying the conditions of Lemma 11. Let and be the same matrices as in the proof of Lemma 16. It can be directly verified that the family of four skew-symmetric matrices satisfies all conditions of Lemma 11, and so it makes skew-symmetric

Corollary 19 (see [1]). Given any sequence of positive integers, there exists a positive integer such that there is

The following theorem, due to Geramita and Seberry, is known.

Theorem 20 (Geramita and Seberry [1, Theorem 7.14]). Suppose that is a square. Then there is an integer such that, for each , there is

We use a slightly different method to the proof of Theorem 20 to give a proof of the following improved result.

Theorem 21. Suppose that is a square. Then there is an integer such that, for each , there is symmetric

Proof. Assume that , where is either 1 or a prime power. By Theorem 14, for each there exists circulant Let where is the back-diagonal matrix of order . It can be seen that is symmetric
Thus, there is an odd number such that there is symmetric Moreover, from Lemma 16, there exists symmetric , and so there is symmetric Now since is odd, Lemma 13 implies that there is a positive integer such that, for each , there exists symmetric

We prove the following theorem by a slightly different method to the proof that first was given by Eades [1, 8].

Theorem 22. Suppose that , where and are two nonzero integers. Then there is an integer such that, for each , there is

Proof. For , let , where is either 1 or a prime power. For each , , let From Corollary 15, for each , , and each , , there exists circulant It can be seen that the following matrix is , where is the back-diagonal matrix of order and is an odd number. From Theorem 17, one can choose the smallest positive integer such that there is Since , Lemma 13 implies that there is an integer such that, for each , there is

Using the methodology in the proof of Theorem 22, the asymptotic bounds for the following two corollaries given by Eades [8] are improved.

Corollary 23. Suppose that is the sum of two nonzero integer squares. Then there is an integer such that, for each , there is

Proof. Let , where and are integers. From Theorem 22, there is an integer such that, for any , there is , and so there is

Corollary 24. Suppose that is an integer square. Then there exists an integer such that, for each , there is skew-symmetric

Proof. Suppose that Let and By Theorem 22, there exists an integer such that, for each , there is , and so there is skew-symmetric

We now use a different method to show Theorem 25 shown by Eades [1, 8] to improve the bounds () for the asymptotic existence of ODs of order , and consequently we prove Corollaries 26, 27, and 28.

Theorem 25. Suppose that , where , and are nonzero integers. Then there is an integer such that, for each , there is

Proof. Assume that and , and are nonzero integers. Let , where is either 1 or a prime power. For each , , let From Corollary 15, for each , , and each , , there exists circulant Putting in the following array (Goethals and Seidel [10]) gives : where which is an odd number and is the back-diagonal matrix of order .
By Lemma 18, there is for some suitable integer . Since, for , , Lemma 13 implies that there is an integer such that, for each , there is Note that if some of ’s are zero, then we consider the zero matrices.

Corollary 26. Suppose that is any positive integer. Then there is an integer such that, for each , there is

Proof. It is a well known theorem of Lagrange [11] that every positive integer can be written in the sum of four integer squares. Let From Theorem 25, there is an integer such that, for each , there is , and therefore there is

Corollary 27. Suppose that is the sum of three integer squares. Then there exists an integer such that, for each , there is skew-symmetric

Proof. Consider , for some integers , and Substituting , , , and in Theorem 25 gives the result. Note that the existence of is equivalent to existence of skew-symmetric

Corollary 28. Suppose that is any positive integer. Then there exists an integer such that, for each , there is skew-symmetric

Proof. By Lagrange’s theorem [12], one can write , where ’s are nonnegative integers. Let , and be the same matrices as in Theorem 25. It can be seen that the following matrix gives , where is obtained as in Theorem 25, and is an odd number: From Corollary 19, there is for some suitable integer Since, for , , Lemma 13 implies that there is an integer such that, for any , there is , and so there is skew-symmetric

Example 29. Suppose that Let , and in Theorem 25. Also, let , and Then , and By Theorem 25, there is From Lemma 18, there is . By Lemma 13, since , we have , and so, for each , there are and skew-symmetric .

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The paper constitutes Chapter 3 of the author’s Ph.D. thesis written under the direction of Professor Hadi Kharaghani at the University of Lethbridge. The author would like to thank Professor Hadi Kharaghani for introducing the problem and his very useful guidance toward solving the problem and also Professor Rob Craigen for his time and great help.