# Constructions of optimal orthogonal arrays with repeated rows

## TL;DR

This paper presents new methods for constructing optimal orthogonal arrays with repeated rows, maximizing the ratio of repeated rows to the array's index, and explores the existence of basic OAs linked to Hadamard matrices.

## Contribution

It introduces constructions for optimal orthogonal arrays with large repeated row ratios and establishes a link between basic OAs with n=2 and Hadamard matrices.

## Key findings

- Constructed optimal OAs for any k ≥ n+1 with large λ
- Developed basic OAs with n=2 and k=4t+1 based on Hadamard matrices
- Solved the problem of constructing basic OAs with n=2 modulo Hadamard conjecture

## Abstract

We construct orthogonal arrays OA$_{\lambda} (k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m / \lambda$ is as large as possible; these OAs are termed optimal. We provide constructions of optimal OAs for any $k \geq n+1$, albeit with large $\lambda$. We also study basic OAs; these are optimal OAs in which $\gcd(m,\lambda) = 1$. We construct a basic OA with $n=2$ and $k =4t+1$, provided that a Hadamard matrix of order $8t+4$ exists. This completely solves the problem of constructing basic OAs wth $n=2$, modulo the Hadamard matrix conjecture.