# The Independence Number of the Orthogonality Graph in Dimension $2^k$

## TL;DR

This paper determines the independence number of the orthogonality graph on hypercubes of dimension 2^k, resolving a long-standing question related to quantum information theory.

## Contribution

It extends Frankl's rank argument to compute the independence number for hypercubes of dimension 2^k, answering a question posed in 2001.

## Key findings

- Exact independence number for the orthogonality graph in dimension 2^k
- Extension of Frankl's method to new hypercube dimensions
- Resolution of a problem in quantum information theory

## Abstract

We determine the independence number of the orthogonality graph on $2^k$-dimensional hypercubes. This answers a question by Galliard from 2001 which is motivated by a problem in quantum information theory. Our method is a modification of a rank argument due to Frankl who showed the analogous result for $4p^k$-dimensional hypercubes, where $p$ is an odd prime.