Symmetric Hadamard matrices of order 116 and 172 exist

TL;DR

This paper constructs new symmetric Hadamard matrices of orders 116 and 172, expanding known existence results, using combinatorial arrays and algorithmic adaptations, and introduces an infinite series of such matrices.

Contribution

It presents the first known symmetric Hadamard matrices of orders 116 and 172, and introduces a new infinite series based on the GP array.

Findings

Existence of symmetric Hadamard matrices of order 116 and 172 confirmed.

New combinatorial array and algorithmic methods applied.

An infinite series of symmetric Hadamard matrices identified.

Abstract

We construct new symmetric Hadamard matrices of orders $92,116$, and $172$. While the existence of those of order $92$ was known since 1978, the orders $116$ and $172$ are new. Our construction is based on a recent new combinatorial array discovered by N. A. Balonin and J. Seberry. For order $116$ we used an adaptation of an algorithm for parallel collision search. The adaptation pertains to the modification of some aspects of the algorithm to make it suitable to solve a 3-way matching problem. We also point out that a new infinite series of symmetric Hadamard matrices arises by plugging into the GP array the matrices constructed by Xia, Xia, Seberry, and Wu in 2005.