# Quaternion Codes in MIMO System of Dual-Polarized Antennas +
## 2 Quaternion Orthogonal Designs
### Maximal Rate QODs for General Configuration of Dual-Polarized Antennas  
In [PERSON]'s paper [3], rectangular CODs of maximal rates are found algorithmically. The general procedure was given in the paper, however, such designs do not carry a compact form as was found in the case of (2). As demonstrated below that our proposed procedure successfully works on the famous examples given in [3], to generate designs for 3 and 5 dual-polarized antennas among other configurations, respectively.  
**Lemma 1**.: _A maximal rate \(\mathcal{R}_{\mathbf{Q}}=3/4\) QOD for 3 dual-polarized antennas is given by_  
\[\mathbf{Q}_{4}=\begin{bmatrix}z_{1}+z_{2}j&z_{2}+z_{1}j&z_{3}+z_{3}j\\ -z_{2}^{*}+z_{1}^{*}j&z_{1}^{*}-z_{2}^{*}j&0\\ -z_{3}^{*}&-z_{3}^{*}j&z_{1}^{*}+z_{1}^{*}j\\ -z_{3}^{*}j&-z_{3}^{*}&z_{2}^{*}+z_{2}^{*}j\end{bmatrix}. \tag{11}\]  
Proof.: We start with a rectangular COD of maximal rate \(3/4\) given by  
\[\mathbf{U}=\begin{bmatrix}z_{1}&z_{2}&z_{3}\\ -z_{2}^{*}&z_{1}^{*}&0\\ -z_{3}^{*}&0&z_{1}^{*}\\ 0&-z_{3}^{*}&z_{2}^{*}\end{bmatrix}, \tag{12}\]  
and construct an equivalent COD \(\mathbf{V}\) based on the same principle of permutation of columns such that \(\mathbf{U}^{H}\mathbf{V}\) is symmetric which can be verified easily. Such an equivalent COD is  
\[\mathbf{V}=\begin{bmatrix}z_{2}&z_{1}&z_{3}\\ z_{1}^{*}&-z_{2}^{*}&0\\ 0&-z_{3}^{*}&z_{1}^{*}\\ -z_{3}^{*}&0&z_{2}^{*}\end{bmatrix}. \tag{13}\]  
Consequently, we obtain a QOD for three dual-polarized antennas of rate \(3/4\) given by \(\mathbf{U}+\mathbf{V}j=\mathbf{Q}_{4}\).  
To complete the discussion, we include one QOD which is suitable for 4 dual-polarized antennas and is obtained using [PERSON]'s approach. This will be compared with the QODs based on recursive approach for the same number of antennas.  
\[\mathbf{Q}_{5}=\begin{bmatrix}z_{1}&z_{1}j&z_{2}+z_{3}j&z_{3}+z_{2}j\\ z_{1}j&z_{1}&z_{4}+z_{5}j&z_{5}+z_{4}j\\ -z_{2}^{*}-z_{4}^{*}j&-z_{4}^{*}-z_{2}^{*}j&z_{1}^{*}&z_{1}^{*}j\\ -z_{3}^{*}-z_{5}^{*}j&-z_{5}^{*}-z_{3}^{*}j&z_{1}^{*}j&z_{1}^{*}\\ -z_{4}+z_{2}j&z_{2}-z_{4}j&z_{6}j&z_{6}\\ -z_{5}^{*}j&-z_{6}^{*}&-z_{3}^{*}+z_{2}^{*}j&z_{2}^{*}-z_{3}^{*}j\\ -z_{5}+z_{3}j&z_{3}-z_{5}j&-z_{6}&-z_{6}\\ z_{6}^{*}&z_{6}^{*}j&-z_{5}^{*}+z_{4}^{*}j&z_{4}^{*}-z_{5}^{*}j\end{bmatrix}. \tag{14}\]Lastly, we construct a QOD for 5 dual-polarized antennas.  
**Lemma 2**.: _A maximal rate \(\mathcal{R}_{\mathbf{Q}}=2/3\) QOD for 5 dual-polarized antennas is given by_  
\[\mathbf{Q}_{6}=\begin{bmatrix}z_{1}&z_{1}j&z_{2}+z_{3}j&z_{3}+z_{2}j&z_{4}+z_{4} j\\ z_{1}j&z_{1}&z_{5}+z_{6}j&z_{6}+z_{5}j&z_{7}+z_{7}j\\ -z_{2}^{*}-z_{5}^{*}j&-z_{5}^{*}-z_{2}^{*}j&z_{1}^{*}&z_{1}^{*}j&0\\ -z_{3}^{*}-z_{6}^{*}j&-z_{6}^{*}-z_{3}^{*}j&z_{1}^{*}j&z_{1}^{*}&0\\ -z_{4}^{*}-z_{7}^{*}j&-z_{7}^{*}-z_{4}^{*}j&0&0&z_{1}^{*}+z_{1}^{*}j\\ -z_{5}+z_{2}j&z_{2}-z_{5}j&z_{8}j&z_{8}&z_{9}+z_{9}j\\ -z_{8}^{*}j&-z_{8}^{*}&-z_{3}^{*}+z_{2}^{*}j&z_{2}^{*}-z_{3}^{*}j&0\\ -z_{9}^{*}j&-z_{9}^{*}&-z_{4}^{*}&-z_{4}^{*}j&z_{2}^{*}+z_{2}^{*}j\\ -z_{6}+z_{3}j&z_{3}-z_{6}j&-z_{8}&-z_{8}j&z_{10}+z_{10}j\\ -z_{10}^{*}j&-z_{10}^{*}&-z_{4}^{*}j&-z_{4}^{*}&z_{3}^{*}+z_{3}^{*}j\\ -z_{7}+z_{4}j&z_{4}-z_{7}j&-z_{9}-z_{10}j&-z_{10}-z_{9}j&0\\ z_{8}^{*}&z_{8}^{*}j&-z_{6}^{*}+z_{5}^{*}j&z_{5}^{*}-z_{6}^{*}j&0\\ z_{9}^{*}&z_{9}^{*}j&-z_{7}^{*}&-z_{7}^{*}&z_{5}^{*}+z_{5}^{*}j\\ z_{10}^{*}&z_{10}^{*}j&-z_{7}^{*}j&-z_{7}^{*}&z_{6}^{*}+z_{6}^{*}j\\ 0&0&z_{10}^{*}-z_{9}^{*}j&-z_{9}^{*}+z_{10}^{*}j&z_{8}^{*}+z_{8}^{*}j\end{bmatrix}. \tag{15}\]  
Proof.: The proof of this lemma is similar to the Lemma 1, however, in this case the underlying COD \(\mathbf{U}\) is given in Equation (100) in [3].  
Following the same lines, it is easy to construct QODs for higher number of transmit antennas \(n_{T}=6,7,8,\dots\) starting with the CODs given in [3] (Equation (101) and Appendices C and D, respectively).