On the Uniqueness of Balanced Complex Orthogonal Design

TL;DR

This paper proves that for a fixed number of columns, all indecomposable balanced complex orthogonal designs are structurally identical and share the same parameters, highlighting their uniqueness.

Contribution

It establishes the uniqueness and equivalence of indecomposable BCODs with fixed columns, clarifying their structure and parameters.

Findings

All indecomposable BCODs with fixed columns have the same parameters.

Such BCODs are all equivalent to each other.

The parameters of these BCODs are [2^m, 2m, 2^{m-1}].

Abstract

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.