Construction of symmetric Hadamard matrices

TL;DR

This paper introduces a new construction method for symmetric Hadamard matrices, providing the first examples at certain orders and identifying the smallest unresolved order, advancing combinatorial matrix theory.

Contribution

It presents the propus construction method for symmetric Hadamard matrices and offers the first known examples at specific orders, notably 156.

Findings

First symmetric Hadamard matrix of order 156 constructed

Smallest unresolved order for symmetric Hadamard matrices identified as v=47

Systematic exploration of the propus construction method

Abstract

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 $156=4\cdot 39$. The smallest $v$ for which symmetric Hadamard matrices of order $4v$ are not known is now $v=47$.