# Quaternion Codes in MIMO System of Dual-Polarized Antennas +
## 2 Quaternion Orthogonal Designs  
A quaternion consists of four real numbers \(q=x_{0}+x_{1}i+x_{2}j+x_{3}k\), such that \(x_{r}\in\mathbb{R}\); \(r=\{0,1,2,3\}\). As the space \(\mathbb{R}^{4}\) is isomorphic to the complex space \(\mathbb{C}\times\mathbb{C}\), therefore, we obtain an equivalent representation of a quaternion \(q=z_{1}+z_{2}j\). This is more useful from the point of wireless communication among dual-polarized antennas. Consequently, it makes sense to talk about space-time-polarization block codes with quaternion elements and investigate orthogonality condition in the quaternion domain. **Definition 1**. : _A QOD \(\mathbf{Q}\) comprises of quaternion variables \(\{q_{1},q_{2},\ldots,q_{u}\}\), is defined as an \(m\times n\) matrix which can have entries from the set \(\{0,\pm q_{1},\pm q_{1}^{Q},\ldots,\pm q_{u},\pm q_{u}^{Q}\}\), including possible multiplications on the left and/or right by quaternion elements and satisfies the following condition_  
\[\mathbf{Q}^{Q}\mathbf{Q}=\sum_{h=1}^{u}(|q_{h}|^{2})\mathbf{I}_{n}=\lambda \mathbf{I}_{n}, \tag{1}\]  
_where \(\lambda\) is a positive real number and \(\mathbf{I}_{n\times n}\) is an \(n\times n\) identity matrix._  
As mentioned before that a quaternion is a combination of two complex numbers \(q=z_{1}+z_{2}j\), therefore, it is natural to think of QODs satisfying \(\mathbf{Q}=\mathbf{A}+\mathbf{B}j\), where \(\mathbf{A}\) and \(\mathbf{B}\) are two complex matrices. It turned out that any two arbitrary complex matrices do not necessarily give rise to a QOD which satisfies (1). Essentially, the authors in [6] found the key requirements on \(\mathbf{A}\) and \(\mathbf{B}\) to ensure (1), for the resulting QOD. Interestingly, the _amicable_ and _symmetry_ conditions were found to play main role for which an extensive theory was already in place and they used it to generate class of QODs. Later it was found that all proposed quaternion-based designs employed symmetric-paired complex matrices. Since the symmetry property is crucial in our study therefore we state it for a brief and self-contained exposition. **Definition 2**. : _Two CODs \(\mathbf{A}\) and \(\mathbf{B}\) based on complex variables \(\{z_{1},z_{2},\ldots,z_{u}\}\) form a symmetric-paired design \((\mathbf{A}+\mathbf{B}j)\) provided \(\mathbf{A}^{H}\mathbf{B}\) or \(\mathbf{B}^{H}\mathbf{A}\) is symmetric._  
A relatively simple way to find such symmetric-paired designs arise from the observation that swapping of certain columns of a COD generates an equivalent COD. The resulting COD along with the original COD form a symmetric-paired design. This technique was used in [6], for the search of viable QODs. However, it is important to note that as the dimension of COD matrix gets larger, not every permutation of columns of a COD yields a valid QOD. Therefore, only one permutation per column is allowed to generate valid QODs under this construction [15]. Below in Sections 2.2 and 2.3, we will describe other possible ways to generate QODs. For square designs, three recursive construction methods to find CODs were presented in [3], namely Adams-Lax-Phillips, Jozefiak and Wolfe constructions. It is easy to realize that all three constructions recursively generate the same class of square QODs. According to these constructions, a recursive COD \(\mathbf{A}\) is designed for \(l+1\) symbols embedded in a square matrix of order \(2^{l}\) such that  
\[\mathbf{A}= \begin{bmatrix}\mathbf{G}_{2^{l-1}}(z_{1},z_{2},\dots,z_{l})&z_{l+1} \mathbf{I}_{2^{l-1}}\\ -z_{l+1}^{*}\mathbf{I}_{2^{l-1}}&\mathbf{G}_{2^{l-1}}^{H}(z_{1},z_{2},\dots,z_ {l})\end{bmatrix} , \tag{2}\]  
where \(\mathbf{G}_{2^{l-1}}(z_{1},z_{2},\dots,z_{l})\) represents a COD of order \(2^{l-1}\times 2^{l-1}\) defined on symbols \(z_{1},z_{2},\dots,z_{l}\) and \(l=\{1,2,3,\dots\}\). For example, for \(l=1\), \(\mathbf{G}_{1}(z_{1})=[z_{1}]\). Taking this as a seed element, higher order CODs such as \(\mathbf{G}_{2}(z_{1},z_{2})\) and \(\mathbf{G}_{4}(z_{1},z_{2},z_{3},z_{4})\) can be constructed [3], recursively.