Structure theorem of square complex orthogonal design

TL;DR

This paper establishes a fundamental structure theorem for square complex orthogonal designs, providing a deeper understanding of their algebraic properties and limitations.

Contribution

It presents a comprehensive structure theorem for square CODs, extending previous bounds and connecting them to group representation theory.

Findings

Proves the structure theorem for square CODs.

Establishes an upper bound on the rate of square CODs.

Links the structure of CODs to group representations.

Abstract

Square COD (complex orthogonal design) with size $[n, n, k]$ is an $n \times n$ matrix $\mathcal{O}_z$, where each entry is a complex linear combination of $z_i$ and their conjugations $z_i^*$, $i=1,\ldots, k$, such that $\mathcal{O}_z^H \mathcal{O}_z = (|z_1|^2 + \ldots + |z_k|^2)I_n$. Closely following the work of Hottinen and Tirkkonen, which proved an upper bound of $k/n$ by making a crucial observation between square COD and group representation, we prove the structure theorem of square COD.