# User-Centric Cell-Free Massive MIMO with Low-Resolution ADCs for Massive Access
## 4 AP-UE Association Algorithm  
In this section, we propose an AP-UE association scheme for user-centric cell-free massive MIMO systems in the context of massive access scenarios in which the number of UEs in the network is significantly greater than the number of APs. When each UE connects to the network, it selects the APs that will support it. To avoid significant pilot contamination, each AP has a limited capacity to support UEs. Also, we apply the following assumptions. **Assumption 1**. : _Each AP serves at most one UE per pilot and uses all \(N\) antennas to serve these UEs._  
\[\mathbf{D}_{k}^{l}= \begin{cases}\mathbf{1}_{N},\text{ if }k\in\mathcal{A}_{l}\\ \mathbf{0}_{N},\text{ if otherwise}\end{cases}  \tag{35}\]  
_for \(l=1,\dots,L\)._  
**Assumption 2**. : _In scenarios where the number of UEs exceeds \(L\times\tau_{p}\), it is assumed that the CPU allocates different time or frequency resources to APs to accommodate the additional UEs. Each allocated resource operates independently, allowing APs to manage their associated UEs without interference from APs operating on different resources. Consequently, within each allocated resource, the algorithm operates under the assumption that the number of UEs does not exceed \(L\times\tau_{p}\), ensuring optimal performance and minimizing pilot contamination._  
In this study, we satisfy Assumptions 1 and 2, and propose an AP-UE association scheme using a heuristic approach. We consider a scenario where all UEs in the network are connected to at least one AP, and the number of UEs exceeds the number of APs. Each AP has a limited capacity to support UEs, so both AP-preference and UE-preference associations need to be considered to ensure that all UEs are connected to at least one AP. The AP-UE association is conducted using LSFC between APs and UEs, enabling connections to APs with higher LSFC. Due to the large number of UEs, there may be some UEs connected to only one AP. These UEs are associated with the APs that provide the largest LSFC possible. Due to Assumption 2, the algorithm effectively operates within the constraint of each resource, making it a solution for AP-UE connections within the limit of \(L\times\tau_{p}\) UEs per resource. The association process operates through the following steps:  
Step 1. During the initial access phase, each UE calculates its LSFC with each AP and selects the AP with the highest LSFC as its master AP, denoted as \(l_{m}\), for connection. Also, all APs have a set of UEs denoted as \(\left\{\mathcal{A}_{l_{m}}^{M}\right\}\), which have selected them as the master AP. As there are more UEs than APs, APs may receive more connection requests than they can support. APs with more connected UEs that they can support exclude the UE with the smallest LSFC from their connections. The excluded UE is included in the _exclusion-list_, denoted as \(\mathcal{E}_{k}\), of that AP. And the excluded UE selects the AP with the highest LSFC among the other APs as its master AP. If the number of UEs exceeds \(L\times\tau_{p}\), the CPU dynamically allocates different time or frequency resources to APs to accommodate the additional UEs. Each allocated resource operates independently, allowing APs to manage their associated UEs without interference from APs operating on different resources. Consequently, within each allocated resource, the algorithm operates under the assumption that the number of UEs does not exceed \(L\times\tau_{p}\), ensuring optimal performance and minimizing pilot contamination. This process continues until there are no over-serving APs. Through this method, connectivity is guaranteed for UEs with APs that can provide good performance, while also preventing any AP from becoming overserved during the initial access process. Step 2. All UEs maintain a list of available AP candidates that exclude the \(\mathcal{A}^{m}\), \(\left\{\mathcal{M}_{k}=\emptyset:k=1,\ldots,K\right\}\) and \(\mathcal{E}_{k}\), which comprises APs capable of supporting them. And the list of AP candidates is represented as \(\mathcal{C}_{k}\).