二次复正交设计的结构定理
Structure theorem of square complex orthogonal design
摘要：
尺寸为$[n, n, k]$的正方形COD（复正交设计）是$n \times n$矩阵$\mathcal{O}_z$，其中每个条目是$z_i$及其共轭$z_i^*$、$i=1,\ldots, k$的复线性组合，使得$\mathcal{O}_z^H \mathcal{O}_z = (|z_1|^2 + \ldots + |z_k|^2)I_n$。紧接着Hottinen和Tirkkonen的工作，通过对平方COD和群表示之间的一个重要观察，证明了$k/n$的一个上界，我们证明了平方COD的结构定理.
英文摘要：
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.
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