# Quaternion Codes in MIMO System of Dual-Polarized Antennas +
###### Abstract  
The use of quaternion orthogonal designs (QODs) to describe point-to-point communication among dual-polarized antennas has the potential to provide higher rate orthogonal and quasi-orthogonal complex designs exploiting polarization diversity among space and time diversities. Furthermore, it is essential to have a space time block code (STBC) which offers a linear and decoupled decoder which quasi-orthogonal designs fail to attain. In this paper, we show how the realm of quaternions unexpectedly offers us a possible solution and codes obtained from quaternion designs mostly achieve both linear and decoupled decoders. This motivated us to perform an indispensable search for QODs such that the code rate is bounded below by 1/2 and does not sharply decrease as the number of transmit antennas increases. It is shown that three famous recursive techniques do not satisfy this criteria and their code rates decrease rather rapidly. Therefore, we propose another method of constructing quaternion designs suitable for any number of transmit antennas and verify that these attain linear and decoupled decoders with the system model based on quaternionic channel. It is shown that such designs outperform others in terms of transmit diversity, code rates and the optimality of the proposed decoder is validated through simulation results. q 2021 11 3131  
coupled decoding issues [4]. To further enhance the capacity of communication systems, other combinations of diversity providing techniques are being investigated. In [5], orthogonally polarized transmissions through both ends of a dual-polarized antenna were modeled through quaternions and later [6] laid the foundation of orthogonal space time polarization block codes (OSTPBCs) that utilize polarization diversity together with space and time diversity. Polarization diversity can provide nearly similar performance to spatial diversity without any measurable increase in antenna dimensions [7]. This is achieved with the use of dual-polarized antennas, which have two antennas of orthogonal polarizations co-existing on a single antenna platform and there has been a growing interest recently [8; 9]. Based on the combination of polarization diversity with space and time diversities, various quaternion orthogonal design (QOD) construction techniques have been proposed by [PERSON] et al. [6]. The primary motivation of these designs has been their ability to provide higher code rates along with a low complexity quaternion norm-based ML decoder. To illustrate the benefit of these designs, [6] presented an example of \(2\times 2\) order QOD and derived linear equation based decoding solution for this configuration. They argued that quaternion decoding statistics can provide decoupled decoding for any QOD. Their subsequent studies [10; 11] used the same QOD and emphasized the similar postulate that quaternion ML norm can provide optimal decoupled decoding for any QOD construction. However, the authors corrected their decoding rule in [12] and highlighted that the proposed decoding rule does not yield optimal decoding for all QODs, and therefore, the design of semi-optimal or optimal low complexity decoders remained an open research problem [13]. In this regard, a recent work [14], explored the designs for which quaternion norm-based ML decoder resulted in optimal decoding solutions. However, it is important to note that their proposed ML decoder works for a special class of STBCs. In this paper, we investigate three famous generalized QOD construction techniques [14] and identify two main short falls which restricts their use for large MIMO systems. Firstly, this iterative approach works only in the case when the number of transmit antennas are in powers of 2, which clearly restricts their use to other antenna configurations. Secondly, the code rate decreases very sharply for higher order designs based on these iterative techniques.