Symmetric Hadamard matrices of orders 268, 412, 436 and 604

TL;DR

This paper reports the first construction of symmetric Hadamard matrices of orders 268, 412, 436, and 604 using the propus construction and difference families with nontrivial multipliers.

Contribution

It introduces a new method for constructing symmetric Hadamard matrices of specific orders by restricting the search to certain difference families.

Findings

Constructed symmetric Hadamard matrices of orders 268, 412, 436, and 604.

Used the propus construction with difference families admitting nontrivial multipliers.

First known constructions of these specific symmetric Hadamard matrices.

Abstract

We construct many symmetric Hadamard matrices of small order by using the so called propus construction. The necessary difference families are constructed by restricting the search to the families which admit a nontrivial multiplier. Our main result is that we have constructed, for the first time, symmetric Hadamard matrices of order 268, 412, 436 and 604.