The Propus Construction for Symmetric Hadamard Matrices

TL;DR

This paper introduces the Propus construction method for symmetric Hadamard matrices, providing new constructions for matrices of certain orders and generalizations of the array to broader classes.

Contribution

The paper presents a novel Propus array construction for symmetric Hadamard matrices, including new methods based on symmetric Williamson-type matrices and generalizations like the GP array.

Findings

Constructed symmetric propus-Hadamard matrices for 57 orders less than 200.

Developed variations of the Propus array for more general matrices.

Extended the Propus construction to include the Generalized Propus Array (GP).

Abstract

\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)}.