**1. Introduction**

The increasing number of multimedia services and online platforms are reflected as a growth in the demand for connectivity and throughput worldwide. According to Cisco [1], in 2018 there were 3.9 billion Internet users globally, and it estimates 5.3 billion in 2023, representing 66% of the global population by that year according to [2]. Part of the global data traffic, 54%, will originate from mobile devices which will be responsible for 160 Exabytes (EB) per month of data traffic in 2025, a 321% growth when compared to the 38 EB per month in 2019 [3].

To support this demand, the wireless communications standard must evolve and increase the total available throughput and the supported number of connected devices, while keeping the average network latency to a minimum. These characteristics will even make new services possible, such as autonomous cars [4].

Massive MIMO is a technology incorporated in 5G networks which corresponds to the transmission of data through several antennas. Thomas L. Marzetta describes the use of this technology for any multi-user MIMO system with more than 16 antennas [5]. The proposed communications scheme relied on low complexity transceivers in the mobile devices and time division duplexing (TDD) to provide a reliable communication channel.

Within a Massive MIMO system, an arbitrary array of bits, known as a pilot sequence, is designated to each user and used in the uplink training process to simplify the signal detection routines in mobile terminals. However, the use of non-orthogonal sequences causes interference among users, which is known as the pilot contamination problem.

**Citation:** Matos, E.A.; Parmezan Bonidia, R.; Sipoli Sanches, D.; Santos Pozza, R.; Dias Hiera Sampaio, L. Pilot Sequence Allocation Schemes in Massive MIMO Systems Using Heuristic Approaches. *Appl. Sci.* **2022**, *12*, 5117. https://doi.org/ 10.3390/app12105117

Academic Editors: Peng-Yeng Yin, Jen-Chun Lee, Hua-Yi Lin, Ming-Chin Chuang, Youcef Gheraibia and Ray-I Chang

Received: 30 March 2022 Accepted: 13 May 2022 Published: 19 May 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

To guarantee orthogonality, the size of the pilot sequence must increase with respect to the number of users in the system. Hence, it is easy to verify that pilot sequences must be reused in multi-cell environments and high-density urban areas to avoid long periods of uplink training since as the number of orthogonal sequences increases the size of these sequences also increases.

This work proposes to solve the pilot sequence allocation problem in multi-cell scenarios in a centralized fashion with the maximization of the network spectral efficiency as a goal. The optimization problem is modeled into two different, but equivalent, ways: a binary optimization problem and an integer optimization problem. Meanwhile, to solve these optimization problems, we used different versions of the Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) metaheuristics.
