# Morphisms of Skew Hadamard Matrices

## TL;DR

This paper introduces a new method to construct real Hadamard matrices from quaternary unit Hadamard matrices and establishes existence and nonexistence results for these matrices.

## Contribution

It presents a novel construction technique for real Hadamard matrices from QUH matrices and provides new existence and nonexistence conditions.

## Key findings

- Constructed real Hadamard matrices of order q^n+q^{n-1} for prime powers q ≡ 3 mod 4
- Established nonexistence conditions for QUH matrices
- Reproduced known existence results through new construction methods

## Abstract

Quaternary unit Hadamard (QUH) matrices were introduced by Fender, Kharagani and Suda along with a method to construct them at prime power orders. We present a novel construction of real Hadamard matrices from QUH matrices. Our construction recovers the result by Mukhopadhyay on the existence of real Hadamard matrices of order $q^n+q^{n-1}$ for each prime power $q\equiv 3\mod 4$. Furthermore we provide nonexistence conditions for QUH matrices.