  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.
