Related papers: New constructions of Hadamard matrices
In this paper we use a design theoretical approach to construct new, previously unknown complex Hadamard matrices. Our methods generalize and extend the earlier results of de la Harpe--Jones and Munemasa--Watatani and offer a theoretical…
Let $G$ be a finite group and let $h$ be a positive integer. A $\text{BH}(G,h)$ matrix is a $G$-invariant $|G|\times |G|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|G|I_{|G|}$, where $H^*$ denotes the complex…
Hadamard matrices are square $n\times n$ matrices whose entries are ones and minus ones and whose rows are orthogonal to each other with respect to the standard scalar product in $\Bbb R^n$. Each Hadamard matrix can be transformed to a…
Using a class of permutation polynomials of $F_{3^{2h+1}}$ obtained from the Ree-Tits symplectic spreads in $PG(3,3^{2h+1})$, we construct a family of skew Hadamard difference sets in the additive group of $F_{3^{2h+1}}$. With the help of a…
We find new constructions of infinite families of skew Hadamard difference sets in elementary abelian groups under the assumption of the existence of cyclotomic strongly regular graphs. Our construction is based on choosing cyclotomic…
A finite sequence of numbers is perfect if it has zero periodic autocorrelation after a nontrivial cyclic shift. In this work, we study quaternionic perfect sequences having a one-to-one correspondence with the binary sequences arising in…
Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the connection between tiling Abelian groups and constructions of complex Hadamard…
A new construction of complex Hadamard matrices of composite order d=pq, with primes p,q, is presented which is based on pairs of mutually unbiased bases containing only product states. For product dimensions d < 100, we illustrate the…
In this paper we disprove the Haagerup statement that all complex Hadamard matrices of order five are equivalent with the Fourier matrix $F_5$ by constructing circulant matrices that lead to new Hadamard matrices. An important item is the…
We construct several difference families on cyclic groups of orders 47 and 97, and use them to construct skew-Hadamard matrices of orders 188 and 388. Such difference families and matrices are constructed here for the first time. The…
In this paper we introduce modular symmetric designs and use them to study the existence of Hadamard matrices modulo 5. We prove that there exist 5-modular Hadamard matrices of order n if and only if n != 3, 7 (mod 10) or n != 6, 11. In…
We introduce a new class of complex Hadamard matrices which have not been studied previously. We use these matrices to construct a new infinite family of parity proofs of the Kochen-Specker theorem. We show that the recently discovered…
We show that 138 odd values of n less than 10000 for which one knows how to construct a Hadamard matrix of order 4n have been overlooked in the recent handbook of combinatorial designs. There are four additional odd n, namely 191, 5767,…
Let $S(x)$ be the number of $n \leq x$ for which a Hadamard matrix of order $n$ exists. Hadamard's conjecture states that $S(x)$ is about $x/4$. From Paley's constructions of Hadamard matrices, we have that \[ S(x) = \Omega(x/\log x). \] In…
In this paper we investigate the structure of the critical groups of doubly regular tournaments (DRTs) associated with skew Hadamard difference families (SDFs) with one, two, or four blocks. Brown and Reid found the existence of a skew…
First examples of symmetric Hadamard matrices of orders 508 and 764 are constructed. The method used is known as the propus construction. A conjecture regarding this method is formally proposed but it appears implicitly in three previous…
Complex Hadamard matrices H of order 6 are characterized in a novel manner, according to the presence/absence of order 2 Hadamard submatrices. It is shown that if there exists one such submatrix, H is equivalent to a Hadamard matrix where…
The Hadamard maximal determinant problem asks for the largest n-by-n determinant with entries in {+1,-1}. When n is congruent to 1 (mod 4), the maximal excess construction of Farmakis & Kounias has been the most successful general method…
The optimal branch number of MDS matrices has established their importance in designing diffusion layers for various block ciphers and hash functions. As a result, numerous matrix structures, including Hadamard and circulant matrices, have…
We systematically explore the new method of construction (known as the propus construction) of symmetric Hadamard matrices for small orders, $4v$. In particular we give the first examples of symmetric Hadamard matrices of order $156=4\cdot…