New constructions of Hadamard matrices

TL;DR

This paper introduces new infinite families of Hadamard matrices constructed via novel difference families and array applications, expanding the known orders and structures of these matrices.

Contribution

The paper presents four new constructions of difference families and applies array techniques to generate previously unknown Hadamard matrices of specific orders.

Findings

New infinite families of Hadamard matrices of orders related to primes and prime powers.

Explicit constructions for matrices of orders 4(uv+1) and 8(uv+1).

Extension of known Hadamard matrix orders using difference families and array methods.

Abstract

In this paper, we obtain a number of new infinite families of Hadamard matrices. Our constructions are based on four new constructions of difference families with four or eight blocks. By applying the Wallis-Whiteman array or the Kharaghani array to the difference families constructed, we obtain new Hadamard matrices of order $4(uv+1)$ for $u=2$ and $v\in \Phi_1\cup \Phi_2 \cup \Phi_3 \cup \Phi_4$; and for $u\in \{3,5\}$ and $v\in \Phi_1\cup \Phi_2 \cup \Phi_3$. Here, $\Phi_1=\{q^2:q\equiv 1\pmod{4}\mbox{ is a prime power}\}$, $\Phi_2=\{n^4\in \mathbb{N}:n\equiv 1\pmod{2}\} \cup \{9n^4\in \mathbb{N}:n\equiv 1\pmod{2}\}$, $\Phi_3=\{5\}$ and $\Phi_4=\{13,37\}$. Moreover, our construction also yields new Hadamard matrices of order $8(uv+1)$ for any $u\in \Phi_1\cup \Phi_2$ and $v\in \Phi_1\cup \Phi_2 \cup \Phi_3$.