On the uniqueness of balanced complex orthogonal design

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 \(2^m\) $2^m$ , 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 \([2^m, 2m, 2^{m-1}]\) $[2^m,2m,2^{m-1}]$ , and moreover, they are all equivalent.