# Optimally Perturbed Identity Matrices of Rank 2

## TL;DR

This paper investigates the tightness of a relaxation technique for optimal antipodal codes, providing exact results for real-valued rank 2 matrices and advancing understanding of low-rank Gram matrix bounds.

## Contribution

It offers new insights into the relaxation's accuracy and derives exact results for rank 2 real-valued matrices in the context of antipodal code optimization.

## Key findings

- Established the tightness of the relaxation for rank 2 matrices
- Derived exact bounds for real-valued rank 2 Gram matrices
- Enhanced understanding of low-rank matrix constraints in coding theory

## Abstract

The problem of optimal antipodal codes can be framed as finding low rank Gram matrices $G$ with $G_{ii} = 1$ and $|G_{ij}| \leq \epsilon$ for $1 \leq i \neq j \leq n$. In 2018, Bukh and Cox introduced a new bounding technique by removing the condition that $G$ be a gram matrix. In this work, we investigate how tight this relaxation is, and find exact results for real valued matrices of rank $2$.