On the existence of Hadamard matrices

Given any natural number q > 3 we show there exists an integer t ? [2log₂(q ? 3)] such that an Hadamard matrix exists for every order 2^sq where s > t. The Hadamard conjecture is that s = 2.This means that for each q there is a finite number of orders 2^υq for which an Hadamard matrix is not known. This is the first time such a statement could be made for arbitrary q.In particular it is already known that an Hadamard matrix exists for each 2^sq where if q = 2^m ? 1 then s ? m, if q = 2^m + 3 (a prime power) then s ? m, if q = 2^m + 1 (a prime power) then s ? m + 1.It is also shown that all orthogonal designs of types (a, b, m ? a ? b) and (a, b), 0 ? a + b ? m, exist in orders m = 2^t and 2^t+2 · 3, t ? 1 a positive integer.     

Complex Hadamard matrices

R. J. Turyn introduced complex Hadamard matrices and showed that if there is a complex Hadamard matrix of order c and a real Hadamard matrix of order h> > 1, then there is a real Hadamard matrix of order he. Previously, complex Hadamard matrices were only known for a few small orders and the orders for which symmetric conference matrices were known. These latter are known only to exist for orders which can be written as 1+a² +b² where a, b are integers. We give many constructions for new infinite classes of complex Hadamard matrices and show that they exist for orders 306,650, 870,1406,2450 and 3782: for the orders 650, 870, 2450 and 3782, a symmetric conference matrix cannot exist.     

A new construction for Williamson-type matrices

It is shown that ifq is a prime power then there are Williamson-type matrices of order

1. 1/2q ²(q + 1) whenq ≡ 1 (mod 4).
2. 1/4q ²(q + 1) whenq ≡ 3 (mod 4) and there are Williamson-type matrices of order 1/4(q + 1).

This gives Williamson-type matrices for the new orders 363, 1183, 1805, 2601, 3174, 5103. The construction can be combined with known results on orthogonal designs to give an Hadamard matrix of the new order 33396 = 4 ? 8349.     

An infinite class of Williamson matrices

Only a finite number of Hadamard matrices of Williamson type are known so far; it has been conjectured that one such exists of any order 4t. An infinite family is constructed here, and as a corollary it is shown that an Hadamard matrix of order 6(q + 1) exists if q is a prime power ≡ 1 (mod 4).     

Regular sets of matrices and applications

SupposeA ₁,...,A _s are (1, - 1) matrices of order m satisfying 1 $$A_i A_j = J, i,j \in \left\{ {1,...s} \right\}$$ 2 $$A_i^T A_j = A_j^T A_i = J, i \ne j, i,j \in \left\{ {1,...,s} \right\}$$ 3 $$\sum\limits_{i = 1}^s {(A_i A_i^T = A_i^T A_i ) = 2smI_m } $$ 4 $$JA_i = A_i J = aJ, i \in \left\{ {1,...,s} \right\}, a constant$$ Call A₁,…,A _s ,a regular s- set of matrices of order m if Eq. 1-3 are satisfied and a regular s-set of regular matrices if Eq. 4 is also satisfied, these matrices were first discovered by J. Seberry and A.L. Whiteman in “New Hadamard matrices and conference matrices obtained via Mathon’s construction”, Graphs and Combinatorics, 4(1988), 355-377. In this paper, we prove that

1. if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn whent =sm
2. if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn when 2t = sm (m is odd)
3. if there exist a regulars-set of order m and a regulart-set of ordern there exists a regular 2s-set of ordermn whent = 2sm As applications, we prove that if there exist a regulars-set of order m there exists
4. an Hadamard matrices of order4hm whenever there exists an Hadamard matrix of order4h ands =2h
5. Williamson type matrices of ordernm whenever there exists Williamson type matrices of ordern and s = 2n
6. anOD(4mp;ms₁,…,ms_u whenever anOD (4p;s₁,…,s_u)exists and s = 2p
7. a complex Hadamard matrix of order 2cm whenever there exists a complex Hadamard matrix of order 2c ands = 2c

This paper extends and improves results of Seberry and Whiteman giving new classes of Hadamard matrices, Williamson type matrices, orthogonal designs and complex Hadamard matrices.     

Skew-Hadamard matrices of order 2(q + 1)

Szekeres has established the existence of a skew-Hadamard matrix of order 2(q + 1) in the case q ≡ 5 (mod 8), a prime power. His method utilized complementary difference sets in the elementary abelian group of order q. The main result of this paper is to show that, for the same q, there exist skew-Hadamard matrices of order 2(q + 1) that are of the Goethals-Seidel type. This is achieved by using a cyclic relative difference set with parameters $\text{(q + 1, 4, q,}\text{1}\text{4}\text{(q ? 1))}$.     

On the asymptotic existence of Hadamard matrices

It is conjectured that Hadamard matrices exist for all orders 4t (t>0). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural numbers k, there is a Hadamard matrix of order k²[a+blog₂k], where a and b are fixed non-negative constants. To prove the Hadamard Conjecture, it is sufficient to show that we may take a=2 and b=0. Since Seberry's ground-breaking result, which showed that we may take a=0 and b=2, there have been several improvements where b has been by stages reduced to 3/8. In this paper, we show that for all ?>0, the set of odd numbers k for which there is a Hadamard matrix of order k²2+[?log₂k] has positive density in the set of natural numbers. The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that there are enough Paley Hadamard matrices to give the result.     

Hadamard matrices of order 4(2p + 1)

It is shown in this paper that if p is a prime and q = 2p ? 1 is a prime power, then there exists an Hadamard matrix of order 4(2p + 1).     

On the asymptotic existence of cocyclic Hadamard matrices

Let q be an odd natural number. We prove there is a cocyclic Hadamard matrix of order ²10+tq whenever . We also show that if the binary expansion of q contains N ones, then there is a cocyclic Hadamard matrix of order ²4N−2q.     

Some notes on integral equivalence of combinatorial matrices

Recent work on integral equivalence of Hadamard matrices and block designs is generalized in two directions. We first determine the two greatest invariants under integral equivalence of the incidence matrix of a symmetric balanced incomplete block design. This enables us to write down all the invariants in the case wherek −λ is square-fre. Some other results on the sequence of invariants are presented. Secondly we consider the existence of inequivalent Hadamard matrices under integral equivalence. We show that if there is a skew-Hadamard matrix of order 8m then there are two inequivalent Hadamard matrices of order 16m, that and there are precisely eleven inequivalent Hadamard matrices of order 32.     

Regular Hadamard Matrices Generating Infinite Families of Symmetric Designs

If H is a regular Hadamard matrix with row sum 2h, m is a positive integer, and q = (2h ? 1)², then (4h ²(q ^{m + 1} ? 1)/(q ?1),(2h ² ? h)q ^m ,(h ²-h)q ^m ) are feasible parameters of a symmetric designs. If q is a prime power, then a balanced generalized weighing matrix BGW((q ^{m +1} ? 1)/(q?1),q ^m ,q ^m ?q ^{m ?1}) can be applied to construct such a design if H satisfies certain structural conditions. We describe such conditions and show that if H satisfies these conditions and B is a regular Hadamard matrix of Bush type, then B×H satisfies these structural conditions. This allows us to construct parametrically new infinite families of symmetric designs.     

Construction of Williamson type matrices

Recent advances in the construction of Hadamard matrices have depeaded on the existence of Baumert-Hall arrays and four (1, ?1) matrices A B C Dof order m which are of Williamson type, that is they pair-wise satisfy

i) MN^T = NM^T , ∈ {A B C D} and

ii) AA^T + BB^T + CC^T + DD^T = 4mI_m .

It is shown that Williamson type matrices exist for the orders m = s(4 ? 1)m = s(4s + 3) for s∈ {1, 3, 5, …, 25} and also for m = 93. This gives Williamson matrices for several new orders including 33, 95,189.

These results mean there are Hadamard matrices of order

i) 4s(4s ?1)t, 20s(4s ? 1)t,s ∈ {1, 3, 5, …, 25};

ii) 4s(4:s + 3)t, 20s(4s + 3)t s ∈ {1, 3, 5, …, 25};

iii) 4.93t, 20.93t

for

t ∈ {1, 3, 5, … , 61} ∪ {1 + 2 ^a 10 ^b 26 ^c a b c nonnegative integers}, which are new infinite families.

Also, it is shown by considering eight-Williamson-type matrices, that there exist Hadamard matrices of order 4(p + 1)(2p + l)r and 4(p + l)(2p + 5)r when p ≡ 1 (mod 4) is a prime power, 8ris the order of a Plotkin array, and, in the second case 2p + 6 is the order of a symmetric Hadamard matrix. These classes are new.     

The cocyclic Hadamard matrices of order less than 40

In this paper all cocyclic Hadamard matrices of order less than 40 are classified. That is, all such Hadamard matrices are explicitly constructed, up to Hadamard equivalence. This represents a significant extension and completion of work by de Launey and Ito. The theory of cocyclic development is discussed, and an algorithm for determining whether a given Hadamard matrix is cocyclic is described. Since all Hadamard matrices of order at most 28 have been classified, this algorithm suffices to classify cocyclic Hadamard matrices of order at most 28. Not even the total numbers of Hadamard matrices of orders 32 and 36 are known. Thus we use a different method to construct all cocyclic Hadamard matrices at these orders. A result of de Launey, Flannery and Horadam on the relationship between cocyclic Hadamard matrices and relative difference sets is used in the classification of cocyclic Hadamard matrices of orders 32 and 36. This is achieved through a complete enumeration and construction of (4t, 2, 4t, 2t)-relative difference sets in the groups of orders 64 and 72.     

On a series of Hadamard matrices of order 2 t and the maximal excess of Hadamard matrices of order 22t

In this paper we give a new series of Hadamard matrices of order 2 ^t . When the order is 16, Hadamard matrices obtained here belong to class II, class V or to class IV of Hall's classification [3]. By combining our matrices with the matrices belonging to class I, class II or class III obtained before, we can say that we have direct construction, namely without resorting to block designs, for all classes of Hadamard matrices of order 16.Furthermore we show that the maximal excess of Hadamard matrices of order 2^2t is 2^3t , which was proved by J. Hammer, R. Levingston and J. Seberry [4]. We believe that our matrices are inequivalent to the matrices used by the above authors. More generally, if there is an Hadamard matrix of order 4n ² with the maximal excess 8n ³, then there exist more than one inequivalent Hadamard matrices of order 2^2t n ² with the maximal excess 2^3t n ³ for anyt 2.     

Orthogonal designs II

Orthogonal designs are a natural generalization of the Baumert-Hall arrays which have been used to construct Hadamard matrices. We continue our investigation of these designs and show that orthogonal designs of type (1,k) and ordern exist for everyk < n whenn = 2 ^t+2?3 andn = 2 ^t+2?5 (wheret is a positive integer). We also find orthogonal designs that exist in every order 2n and others that exist in every order 4n. Coupled with some results of earlier work, this means that theweighing matrix conjecture ‘For every ordern ≡ 0 (mod 4) there is, for eachk ?n, a square {0, 1, ? 1} matrixW = W(n, k) satisfyingWW ^t =kIn’ is resolved in the affirmative for all ordersn = 2^t+1?3,n = 2^t+1?5 (t a positive integer). The fact that the matrices we find are skew-symmetric for allk < n whenn ≡ 0 (mod 8) and because of other considerations we pose three other conjectures about weighing matrices having additional structure and resolve these conjectures affirmatively in a few cases. In an appendix we give a table of the known results for orders ? 64.     

On the asymptotic existence of complex Hadamard matrices

Let N = N(q) be the number of nonzero digits in the binary expansion of the odd integer q. A construction method is presented which produces, among other results, a block circulant complex Hadamard matrix of order 2^αq, where α ≥ 2N - 1. This improves a recent result of Craigen regarding the asymptotic existence of Hadamard matrices. We also present a method that gives complex orthogonal designs of order 2^α+1q from complex orthogonal designs of order 2^α. We also demonstrate the existence of a block circulant complex Hadamard matrix of order 2^βq, where © 1997 John Wiley & Sons, Inc. J Combin Designs 5:319–327, 1997     

Non-skew symmetric orthogonal matrices with constant diagonals

A matrix C of order n is orthogonal if CC^T=dI. In this paper, we restrict the study to orthogonal matrices with a constant m > 1 on the diagonal and ±1's off the diagonal. It is observed that all skew symmetric orthogonal matrices of this type are constructed from skew symmetric Hadamard matrices and vice versa. Some simple necessary conditions for the existence of non-skew orthogonal matrices are derived. Two basic construction techniques for non-skew orthogonal matrices are given. Several families of non-skew orthogonal matrices are constructed by applying the basic techniques to well-known combinatorial objects like balanced incomplete block designs. It is also shown that if m is even and n=0 (mod 4), then an orthogonal matrix must be skew symmetric. The structure of a non-skew orthogonal matrix in the special case of m odd,n=2 (mod 4) and m?1/6n is also studied in detail. Finally, a list of cases with n?50 is given where the existence of non-skew orthogonal matrices are unknown.     

A new construction for Hadamard matrices of orders 8t + 4

In Hadamard matrices of orders 8t + 4, there are usually four rows which agree on exactly one column. In fact, for t = 0, 1, 2 such a “Hall set” always occurs. This is obvious for t = 0, 1 and Hall has shown this for t = 2. When t = 3, the evidence indicates that nearly all H(28) have a Hall set. (Nearly the opposite seems to be true for matrices H(8t).) If a Hall set is assumed to exist for some H and some t, the remaining rows fall into 4 sets which determine 16 submatrices of order t. Several well-known techniques may be applied to such a configuration, and give immediate examples for t = 1, 2, 3, 4.     

Commutativity of the adjacency matrices of graphs

We say that two graphs G₁ and G₂ with the same vertex set commute if their adjacency matrices commute. In this paper, we find all integers n such that the complete bipartite graph K_n,n is decomposable into commuting perfect matchings or commuting Hamilton cycles. We show that there are at most n−1 linearly independent commuting adjacency matrices of size n; and if this bound occurs, then there exists a Hadamard matrix of order n. Finally, we determine the centralizers of some families of graphs.