# A new family of Hadamard matrices of order $4(2q^2+1)$

## TL;DR

This paper constructs a new family of Hadamard matrices of order $4(2q^2+1)$ for specific prime powers $q$, using difference families and the Wallis-Whiteman array, expanding known Hadamard matrix orders.

## Contribution

The paper introduces a novel construction of Hadamard matrices of order $4(2q^2+1)$ for prime powers $q=12c^2+4c+3$, using difference families in finite groups.

## Key findings

- Constructed difference family with specific parameters in ${
m Z}_2 imes {
m F}_{q^2}$.
- Derived Hadamard matrices of order $4(2q^2+1)$ for the given $q$.
- Expanded the class of known Hadamard matrices with new orders.

## Abstract

Let $q$ be a prime power of the form $q=12c^2+4c+3$ with $c$ an arbitrary integer. In this paper we construct a difference family with parameters $(2q^2;q^2,q^2,q^2,q^2-1;2q^2-2)$ in ${\mathbb Z}_2\times ({\mathbb F}_{q^2},+)$. As a consequence, by applying the Wallis-Whiteman array, we obtain Hadamard matrices of order $4(2q^2+1)$ for the aforementioned $q$'s.