# Quaternion Codes in MIMO System of Dual-Polarized Antennas +
## 2 Quaternion Orthogonal Designs
### Symmetric-Paired Design 3: (Non-Square QODs)  
We now come to address main issues related with quasi-CODs which promise a high data rate at the expense of coding and decoding delays besides coupled and non-linear decoding. This issue can be resolved in the quaternion domain. For example, in [16], a quasi-COD of rate 1 was designed for four transmit antennas  
\[\mathbf{C}_{q}= \begin{bmatrix}\mathbf{G}_{2}&\mathbf{L}_{2}\\ -\mathbf{L}_{2}^{*}&\mathbf{G}_{2}^{*}\end{bmatrix} , \tag{7}\]  
using two CODs \(\mathbf{G}_{2}= \begin{bmatrix}z_{1}&z_{2}\\ -z_{2}^{*}&z_{1}\end{bmatrix} \) and \(\mathbf{L}_{2}= \begin{bmatrix}z_{3}&z_{4}\\ -z_{4}^{*}&z_{3}\end{bmatrix} \), which was shown to have pairwise decoding. We now subject these two CODs \(\mathbf{G}_{2}\) and \(\mathbf{L}_{2}\) to the quaternion domain which helps us reclaim decoupled and linear decoding solutions. First of all, we note that the concept used in design 2, can be generalized such that in place of a diagonal matrix which merely contains a single extra symbol, we can use a COD. It turns out that the following theorem provides us a possibility of incorporating \(\mathbf{L}_{2}\) with \(\mathbf{G}_{2}\), in place of a diagonal matrix. Resultantly, we obtain another recursive construction technique which can be proved easily. **Theorem 3**. : _For two recursively generated CODs \(\mathbf{G}_{2^{l-1}}(z_{1},z_{2},\ldots,z_{l})\) and \(\mathbf{L}_{2^{l-1}}(z_{1+2},z_{2},\ldots,\)\(z_{2l+2})\), a symmetric-paired design,_  
\[\mathbf{Q}_{2^{l+1}\times 2^{l}}(z_{1},\ldots,z_{2(l+1)})= \begin{bmatrix} \mathbf{G}_{2^{l}}+\mathbf{L}_{2^{l}}j\\ \mathbf{L}_{2^{l}}+\mathbf{G}_{2^{l}}j\end{bmatrix} , \tag{8}\]  
_is a QOD of dimension \(2^{l+1}\times 2^{l}\)._  
It is easy to implement this technique as is shown in the subsequent example. **Example 3**. : _To generate a QOD for 4 dual-polarized antenna, we use \(\mathbf{G}_{2}\) and \(\mathbf{L}_{2}\) to obtain two CODs of higher orders \(\mathbf{G}_{4}\) and \(\mathbf{L}_{4}\) using Wolfe construction. Consequently, Equation (8) gives rise to a QOD of rate \(3/4\)_  
\[\mathbf{Q}_{3}= \begin{bmatrix}z_{1}+z_{4}j&z_{2}+z_{5}j&z_{3}+z_{6}j&0\\ -z_{2}^{*}-z_{5}^{*}j&z_{1}^{*}+z_{4}^{*}j&0&z_{3}+z_{6}j\\ -z_{3}^{*}-z_{6}^{*}j&0&z_{1}^{*}+z_{4}^{*}j&-z_{2}-z_{5}j\\ 0&-z_{3}^{*}-z_{6}^{*}j&z_{2}^{*}+z_{5}^{*}j&z_{1}+z_{4}j\\ z_{4}+z_{1}j&z_{5}+z_{2}j&z_{6}+z_{3}j&0\\ -z_{6}^{*}-z_{3}^{*}j&z_{4}^{*}+z_{1}^{*}j&0&z_{6}+z_{3}j\\ -z_{6}^{*}-z_{3}^{*}j&0&z_{4}^{*}+z_{1}^{*}j&-z_{5}-z_{2}j\\ 0&-z_{6}^{*}-z_{3}^{*}j&z_{5}^{*}+z_{2}^{*}j&z_{4}+z_{1}j\end{bmatrix} . \tag{9}\]  
Therefore, we have three recursive techniques to generate square and rectangular QODs from square CODs. As all of the above QODs are obtained from the recursive techniques of CODs based on famous Adams-Lax-Phillips, Jozefiak and Wolfe constructions, therefore, we can use an upper bound on these to arrive at the following result. **Theorem 4**. : _The rate \(r_{\mathbf{Q}}\), of all possible QODs in (3), (8) and (11) obtained from square CODs is given by_  
\[r_{\mathbf{Q}}=\frac{l+1}{2^{l}}. \tag{10}\]  
This provides us a class of QODs, which are fully diverse [17] and earlier these were shown to have decoupled decoders based on a semi-quaternionic channel model [15]. In this paper we use the system model [18] based on the characterization of pure quaternionic channel and it is shown that the above designs all have decoupled decoders and optimaldecoding delays. Besides these advantages, there is one drawback as mentioned in the remark below. **Remark 1**.