arXiv subjects

Jennifer Seberry

Publications and source records attributed to Jennifer Seberry.

5 recordsLinked to original sources

arXiv2015

The Propus Construction for Symmetric Hadamard Matrices

\textit{Propus} (which means twins) is a construction method for orthogonal $\pm 1$ matrices based on a variation of the Williamson array called the \textit{propus array} \[ \begin{matrix*}[r] A& B & B & D B& D & -A &-B B& -A & -D & B D& -B & B &-A. \end{matrix*} \] This construction designed to find symmetric Hadamard matrices was originally based on circulant symmetric $\pm 1$ matrices, called \textit{propus matrices}. We also give another construction based on symmetric Williamson-type matrices. We give constructions to find symmetric propus-Hadamard matrices for 57 orders $4n$, $n < 200$ odd. We give variations of the above array to allow for more general matrices than symmetric Williamson propus matrices. One such is the \textit{ Generalized Propus Array (GP)}.

math.CO↗

arXiv2015

On Generalized Hadamard Matrices and Difference Matrices: $Z_6$

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.

math.CO↗

arXiv2015

Cretan(4t+1) Matrices

A $Cretan(4t+1)$ matrix, of order $4t+1$, is an orthogonal matrix whose elements have moduli $\leq 1$. The only $Cretan(4t+1)$ matrices previously published are for orders 5, 9, 13, 17 and 37. This paper gives infinitely many new $Cretan(4t+1)$ matrices constructed using $regular~Hadamard$ matrices, $SBIBD(4t+1,k,λ)$, weighing matrices, generalized Hadamard matrices and the Kronecker product. We introduce an inequality for the radius and give a construction for a Cretan matrix for every order $n \geq 3$.

math.CO↗

arXiv2015

Two-level Cretan Matrices Constructed Theoretically and Computationally using SBIBD

Cretan matrices are orthogonal matrices with elements $\leq 1$. These may have application in forming some new materials. There is a search for Cretan matrices, especially with high determinant, for all orders. These have been found by both mathematical and computational methods. This paper highlights the differences between theoretical and computational solutions to finding Cretan matrices. It has been shown that the incidence matrix of a symmetric balanced incomplete block design can be used to form Cretan($v;2$) matrices. We give families of Cretan matrices constructed using Hadamard related difference sets.

math.CO↗

arXiv2015

Equivalence of the Existence of Hadamard Matrices and Cretan$(4t-1,2)$-Mersenne Matrices

We study orthogonal matrices whose elements have moduli $\leq 1$. This paper shows that the existence of two such families of matrices is equivalent. Specifically we show that the existence of an Hadamard matrix of order $4t$ is equivalent to the existence of a Cretan$(4t-1,2)$-Mersenne matrix.

math.CO↗