Complex orthogonal designs (CODs) play a crucial role in the construction of space-time block codes. Their real analog, real orthogonal designs (or equivalently, sum of squares composition formula) have a long history. Adams et al. (2011) introduced the concept of balanced complex orthogonal designs (BCODs) to address practical considerations. BCODs have a constant code rate of $1/2$ and a minimum decoding delay of $2^m$, where $2m$ is the number of columns. Understanding the structure of BCODs helps design space-time block codes, and it is also fascinating in its own right. We prove, when the number of columns is fixed, all (indecomposable) balanced complex orthogonal designs (BCODs) have the same parameters $[2^m, 2m, 2^{m-1}]$, and moreover, they are all equivalent.

翻译：复正交设计（COD）在空时分组码的构建中起着关键作用。其实数模拟形式，即实正交设计（等价于平方和组合公式），具有悠久的历史。Adams等人（2011）为应对实际需求，引入了平衡复正交设计（BCOD）的概念。BCOD具有恒定的码率 $1/2$ 以及最小解码延迟 $2^m$，其中 $2m$ 为列数。理解BCOD的结构有助于设计空时分组码，其本身也具有重要的研究价值。我们证明，当列数固定时，所有（不可分解的）平衡复正交设计（BCOD）都具有相同的参数 $[2^m, 2m, 2^{m-1}]$，并且，它们彼此等价。