Complex orthogonal designs (CODs) have been used to construct space-time block codes. Its real analog, real orthogonal designs, or equivalently, sum of squares composition formula, have a long history in mathematics. Driven by some practical considerations, Adams et al. (IEEE Trans Info Theory, 57(4):2254–2262, 2011) introduced the definition of balanced complex orthogonal designs (BCODs). The code rate of BCODs is 1/2, and their minimum decoding delay is proven to be \(2^m\), where 2m is the number of columns. 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.

复杂正交设计（COD）已被用来构造时空分组码。它的实模拟、实正交设计，或者等效的平方和合成公式，在数学中有着悠久的历史。出于一些实际考虑，Adams 等人。 （IEEE Trans Info Theory, 57(4):2254–2262, 2011）引入了平衡复数正交设计（BCOD）的定义。 BCOD 的码率为 1/2，其最小解码延迟被证明为\(2^m\) ，其中 2 m是列数。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\([2^m, 2m, 2^{m-1}]\) ，而且，它们都是等价的。