Structure theorem of square complex orthogonal design

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 $z_i^{\ast }$, $i=1,\dots ,k$, such that ${\mathcal{O}}_z^H{\mathcal{O}}_z=(|z_1{|}^2+\dots +|z_k{|}^2)I_n$. 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.