An elementary construction yields a new class of circulant (so‐called “Butson‐type”) generalized weighing matrices, which have order Nn and weight n2, all of whose entries are nth roots of unity, for all positive integers n,N , where n≤N . The idea is extended to a wider class of constructions giving various group‐developed generalized weighing matrices.

It is well-known that a complex Hadamard matrix of order \(2n\) can be used to construct a Hadamard matrix of order \(4n\). Let \(\gamma \) be a primitive complex cube root of unity. In this paper, we describe a method for obtaining a Hadamard matrix of order \(4n\) from a Butson-type generalized Hadamard matrix \(BH(n,6)\) whose entries are drawn from the set \(\{\pm \gamma ,\pm \gamma ^2\}\) of non-real complex sixth roots of unity. We denote such a matrix by \(BH(n,6)\). No \(BH(n,6)\) can exist for odd order \(n\) whose squarefree part is divisible by a prime \(p\equiv 2\pmod 3\). We exhibit examples of such “unreal” \(BH(n,6)\) for all orders \(n<19\) not ruled out by the above condition. We obtain unreal \(BH(n,6)\)’s for \(n=3^{k}hm\), where \(k\) is a nonnegative integer, \(h\) is the order of a Hadamard matrix, and \(m \in \{1,7,10,13\}\).

We give some very interesting matrices which are orthogonal over groups and, as far as we know, referenced, but in fact undocumented. This note is not intended to be published but available for archival reasons.

It is well-known that a complex Hadamard matrix of order 2n can be used to construct a Hadamard matrix of order 4n . Let γ be a primitive complex cube root of unity. In this paper, we describe a method for obtaining a Hadamard matrix of order 4n from a Butson-type generalized Hadamard matrix BH(n,6) whose entries are drawn from the set {±γ ,±γ ^2} of non-real complex sixth roots of unity. We denote such a matrix by BH(n,6) . No BH(n,6) can exist for odd order n whose squarefree part is divisible by a prime p≡ 2 3 . We exhibit examples of such “unreal” BH(n,6) for all orders n<19 not ruled out by the above condition. We obtain unreal BH(n,6) ’s for n=3^khm , where k is a nonnegative integer, h is the order of a Hadamard matrix, and m ∈{1,7,10,13} .

AbstractMotivated by mathematical aspects of origami, Erik Demaine asked which points in the plane can be constructed by using lines whose angles are multiples of $\pi /n$ for some fixed $n$. This has been answered for some specific small values of $n$ including $n=3,4,5,6,8,10,12,24$. We answer this question for arbitrary $n$. The set of points is a subring of the complex plane $\mathbf {C}$, lying inside the cyclotomic field of $n$th roots of unity; the precise description of the ring depends on whether $n$is prime or composite. The techniques apply in more general situations, for example, infinite sets of angles, or more general constructions of subsets of the plane.

Combinatorial design theory is a source of simply stated, concrete, yet difficult discrete problems, with the Hadamard conjecture being a prime example. It has become clear that many of these problems are essentially algebraic in nature. This book provides a unified vision of the algebraic themes which have developed so far in design theory. These include the applications in design theory of matrix algebra, the automorphism group and its regular subgroups, the composition of smaller designs to make larger designs, and the connection between designs with regular group actions and solutions to group ring equations. Everything is explained at an elementary level in terms of orthogonality sets and pairwise combinatorial designs--new and simple combinatorial notions which cover many of the commonly studied designs. Particular attention is paid to how the main themes apply in the important new context of cocyclic development. Indeed, this book contains a comprehensive account of cocyclic Hadamard matrices. The book was written to inspire researchers, ranging from the expert to the beginning student, in algebra or design theory, to investigate the fundamental algebraic problems posed by combinatorial design theory.