Some new symmetric Hadamard matrices

TL;DR

This paper constructs the first known symmetric Hadamard matrices of orders 508 and 764 using the propus method, verifies a related conjecture for several cases, and discusses its implications.

Contribution

It presents the first symmetric Hadamard matrices of orders 508 and 764 and verifies a conjecture for additional cases using the propus construction method.

Findings

Constructed symmetric Hadamard matrices of orders 508 and 764.

Verified the conjecture for v=55,57,59,61,63.

Demonstrated the effectiveness of the propus construction method.

Abstract

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 joint papers with N. A. Balonin et alias. The conjecture has been verified for Hadamard matrices of order 4v with odd v<54. In this note the conjecture is verified for v=55,57,59,61,63.