Hadamard matrices of order 28m, 36m and 44m

We show that if four suitable matrices of order m exist then there are Hadamard matrices of order 28m, 36m, and 44m. In particular we show that Hadamard matrices of orders 14(q + 1), 18(q + 1), and 22(q + 1) exist when q is a prime power and q ≡ 1 (mod 4).Also we show that if n is the order of a conference matrix there is an Hadamard matrix of order 4mn.As a consequence there are Hadamard matrices of the following orders less than 4000: 476, 532, 836, 1036, 1012, 1100, 1148, 1276, 1364, 1372, 1476, 1672, 1836, 2024, 2052, 2156, 2212, 2380, 2484, 2508, 2548, 2716, 3036, 3476, 3892.All these orders seem to be new.     

Automorphisms of higher‐dimensional Hadamard matrices

This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n‐dimensional Hadamard matrix P⁽ⁿ⁾(H). We apply our ideas to the matrices P⁽ⁿ⁾(H). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to cν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P⁽ⁿ⁾(H) is equivalent to “P⁽ⁿ⁾(H′) whenever H is equivalent to H′.” We also show how the automorphism group of P⁽ⁿ⁾(H) depends on the structure of the automorphism group of H. As an application of the above ideas, we determine the automorphism group of P⁽ⁿ⁾(Hk) when Hk is a Sylvester Hadamard matrix of order 2^k. For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P⁽ⁿ⁾(H) where H is equivalent to H₂. These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008     

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.     

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.     

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.     

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.     

On Circulant Complex Hadamard Matrices

We show that a circulant complex Hadamard matrix of order n is equivalent to a relative difference set in the group C ₄×C _n where the forbidden subgroup is the unique subgroup of order two which is contained in the C ₄ component. We obtain non-existence results for these relative difference sets. Our results are sufficient to prove there are no circulant complex Hadamard matrices for many orders.     

A sign test for detecting the equivalence of Sylvester Hadamard matrices

In this paper we show that every matrix in the class of Sylvester Hadamard matrices of order 2 ^k under H-equivalence can have full row and column sign spectrum, meaning that tabulating the numbers of sign interchanges along any row (or column) gives all integers 0,1,...,2 ^k  − 1 in some order. The construction and properties of Yates Hadamard matrices are presented and is established their equivalence with the Sylvester Hadamard matrices of the same order. Finally, is proved that every normalized Hadamard matrix has full column or row sign spectrum if and only if is H-equivalent to a Sylvester Hadamard matrix. This provides us with an efficient criterion identifying the equivalence of Sylvester Hadamard matrices.     

A Hadamard matrix of order 268

The existence of Hadamard matrices of order 268 is established. More generally, suppose that there exist Williamson matrices of orderr. It is shown that this implies the existence of a Hadamard matrix of order 268r. The existence of Baumert-Hall arrays of order 335, and 603 is established as well.     

Robust Hadamard Matrices,Unistochastic Rays in Birkhoff Polytope and Equi-Entangled Bases in Composite Spaces

We study a special class of (real or complex) robust Hadamard matrices, distinguished by the property that their projection onto a 2-dimensional subspace forms a Hadamard matrix. It is shown that such a matrix of order n exists, if there exists a skew Hadamard matrix or a symmetric conference matrix of this size. This is the case for any even \(n\le 20\), and for these dimensions we demonstrate that a bistochastic matrix B located at any ray of the Birkhoff polytope, (which joins the center of this body with any permutation matrix), is unistochastic. An explicit form of the corresponding unitary matrix U, such that \(B_{ij}=|U_{ij}|^2\), is determined by a robust Hadamard matrix. These unitary matrices allow us to construct a family of orthogonal bases in the composed Hilbert space of order \(n \times n\). Each basis consists of vectors with the same degree of entanglement and the constructed family interpolates between the product basis and the maximally entangled basis. In the case \(n=4\) we study geometry of the set \({\mathcal U}_4\) of unistochastic matrices, conjecture that this set is star-shaped and estimate its relative volume in the Birkhoff polytope \({\mathcal B}_4\).     

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.     

Bounds on the number of Hadamard designs of even order

A new lower bound on the number of non‐isomorphic Hadamard symmetric designs of even order is proved. The new bound improves the bound on the number of Hadamard designs of order 2n given in [12] by a factor of 8n ? 1 for every odd n > 1, and for every even n such that 4n ? 1 > 7 is a prime. For orders 8, 10, and 12, the number of non‐isomorphic Hadamard designs is shown to be at least 22,478,260, 1.31 × 10¹⁵, and 10²⁷, respectively. For orders 2n = 14, 16, 18 and 20, a lower bound of (4n ? 1)! is proved. It is conjectured that (4n ? 1)! is a lower bound for all orders 2n ≥ 14. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 363‐378, 2001     

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     

Weaving hadamard matrices with maximum excess and classes with small excess

Weaving is a matrix construction developed in 1990 for the purpose of obtaining new weighing matrices. Hadamard matrices obtained by weaving have the same orders as those obtained using the Kronecker product, but weaving affords greater control over the internal structure of matrices constructed, leading to many new Hadamard equivalence classes among these known orders. It is known that different classes of Hadamard matrices may have different maximum excess. We explain why those classes with smaller excess may be of interest, apply the method of weaving to explore this question, and obtain constructions for new Hadamard matrices with maximum excess in their respective classes. With this method, we are also able to construct Hadamard matrices of near‐maximal excess with ease, in orders too large for other by‐hand constructions to be of much value. We obtain new lower bounds for the maximum excess among Hadamard matrices in some orders by constructing candidates for the largest excess. For example, we construct a Hadamard matrix with excess 1408 in order 128, larger than all previously known values. We obtain classes of Hadamard matrices of order 96 with maximum excess 912 and 920, which demonstrates that the maximum excess for classes of that order may assume at least three different values. Since the excess of a woven Hadamard matrix is determined by the row sums of the matrices used to weave it, we also investigate the properties of row sums of Hadamard matrices and give lists of them in small orders. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 233–255, 2004.     

The structure of weighing matrices having large weights

We examine the structure of weighing matricesW(n, w), wherew=n–2,n–3,n–4, obtaining analogues of some useful results known for the casen–1. In this setting we find some natural applications for the theory ofsigned groups and orthogonal matrices with entries from signed groups, as developed in [3]. We construct some new series of Hadamard matrices from weighing matrices, including the following:W(n, n–2) implies an Hadamard matrix of order2n ifn0 mod 4 and order 4n otherwise;W(n, n–3) implies an Hadamard matrix of order 8n; in certain cases,W(n, n–4) implies an Hadamard matrix of order 16n. We explicitly derive 117 new Hadamard matrices of order 2 ^t p, p<4000, the smallest of which is of order 2³·419.Supported by an NSERC grant     

The excess of complex Hadamard matrices

A complex Hadamard matrix,C, of ordern has elements 1, –1,i, –i and satisfiesCC ^*=nI_nwhereC ^* denotes the conjugate transpose ofC. LetC=[c _ij] be a complex Hadamard matrix of order is called the sum ofC. (C)=|S(C)| is called the excess ofC. We study the excess of complex Hadamard matrices. As an application many real Hadamard matrices of large and maximal excess are obtained.Supported by an NSERC grant.Supported by Telecom grant 7027, an ATERB and ARC grant # A48830241.     

A note on complex matrices that are unitarily congruent to real matrices

Every square complex matrix is known to be consimilar to a real matrix. Unitary congruence is a particular type of consimilarity. We prove that a matrix A∈M_n(C) is unitarily congruent to a real matrix if and only if A is unitarily congruent to via a symmetric unitary matrix. It is shown by an example that there exist matrices that are congruent, but not unitarily congruent, to real matrices.     

Spin models constructed from Hadamard matrices

A spin model (for link invariants) is a square matrix W which satisfies certain axioms. For a spin model W, it is known that W ^T W ^?1 is a permutation matrix, and its order is called the index of W. Jaeger and Nomura found spin models of index?2, by modifying the construction of symmetric spin models from Hadamard matrices. The aim of this paper is to give a construction of spin models of an arbitrary even index from any Hadamard matrix. In particular, we show that our spin models of indices a power of 2 are new.     

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.