Partitioning Hadamard vectors into Hadamard matrices

TL;DR

This paper proves that in a space of dimension m, a family of 2^{m-1} Hadamard vectors can be partitioned into Hadamard matrices only when m is a power of two, providing a simple algorithm for the partitioning.

Contribution

It establishes a necessary and sufficient condition for partitioning Hadamard vectors into matrices and introduces a straightforward algorithm for this process.

Findings

Partitioning is possible only when m is a power of two.

A simple algorithm effectively assigns vectors to Hadamard matrices.

The result characterizes the structure of Hadamard vectors in high-dimensional spaces.

Abstract

We will show that in a space of dimension $m$, any family of $2^{m-1}$ distinct Hadamard vectors (where you can choose x or -x but not both) can be partitioned into Hadamard matrices if and only if $m=2^n$ for some n. We will solve this problem with a simple algorithm for assigning the vectors to the Hadamard matrices.