Hadamard Hypercubes

TL;DR

This paper introduces two novel methods for constructing higher-dimensional Hadamard hypercubes, expanding the understanding of their structure and applications beyond traditional matrices.

Contribution

It presents two new constructions of Hadamard hypercubes, one using conference matrices and the other recursive with Latin hypercubes, linking to association schemes and symmetric designs.

Findings

First construction based on conference matrices and association schemes

Recursive construction combining smaller Hadamard hypercubes and Latin hypercubes

Applications to higher-dimensional symmetric designs

Abstract

Although Hadamard matrices have been investigated since the nineteenth century, relatively little is known about their higher-dimensional analogues. In this paper, we introduce two constructions of Hadamard hypercubes. The first construction is derived from conference matrices, while the second is recursive, combining Hadamard matrices (and hypercubes) of smaller order with Latin hypercubes. The former approach draws on the theory of association schemes on triples, whereas the latter yields applications to the construction of higher-dimensional symmetric designs.