## 1 Introduction  
Low-density parity-check (LDPC) codes [1] have been widely applied to diverse applications, such as wireless communication and data storage systems [2, 3], due to their capability of approaching the capacity under iterative message passing decoding [4]. Many researchers have devoted to developing efficient LDPC decoders [5, 6, 7] to achieve a trade-off between the error rate performance and decoding complexity. Recently, a class of finite alphabet iterative decoders (FAIDs) [8, 9, 10, 11, 12, 13, 14, 15, 16] have drawn much attention due to their excellent performance by using coarsely quantized messages. Due to the use of messages quantized by a low bit width, these FAIDs also achieve a low decoding complexity and are in favor of services and applications such as the Internet of things [17] and wireless sensor networks [18, 19, 20], which require strict power constraint for the devices. Different from the conventional LDPC decoders such as [5], these FAIDs exchange the messages represented by symbols from finite alphabets between the variable nodes (VNs) and the check nodes (CNs). Moreover, they utilize lookup tables (LUTs) with single input to carry out the node updates. These LUTs are carefully designed based on the density evolution (DE) [4] with a selected coarse quantization scheme, which aims to maximize the mutual information (MI) between the coded bits and the exchanged messages within the decoders. We hereby call this type of FAIDs the mutual-information-maximizing FAIDs (MIM-FAIDs). More specifically, the MIM-FAIDs [8, 9, 10, 12, 14, 15] implement the coarse quantization scheme by dynamic programming (DP) [21], which has been proved to be optimal with respect to maximizing MI. In [11; 13], the LUTs of the MIM-FAIDs are designed based on the information bottleneck (IB) method, which makes use of machine learning rather than DP in the design process. Furthermore, the MIM-FAID proposed in [16] constructs the LUTs by using a hierarchical dynamic quantization which is a greedy quantization scheme similar to the IB method and requires less computational complexity compared to DP. In addition, there are two different node updating architectures considered by the above MIM-FAIDs. One is designing multiple sets of concatenated two-input LUTs for decoding, e.g., [8; 11; 12], where each set of LUTs is dedicated to updating the nodes of a specific degree at each iteration. However, the concatenated LUTs only focus on maximizing MI between two consecutive tables after quantization and hence lead to a loss of MI, which may deteriorate the decoder performance. Moreover, the memory requirement for storing the LUTs may be intolerable because the number of LUTs increases significantly when the node degree or the decoding iteration becomes large. To reduce the memory demand and preserve more MI after quantization, the MIM-FAIDs were proposed in [9; 10; 14; 15; 16], which performed the node updates following a reconstruction-calculation-quantization architecture. Specifically, the FAID in [16] utilizes real additions and multiple sets of single-input LUTs with real-valued entries to update all nodes of different degrees. The mutual-information-maximizing (MIM) quantized decoders in [9; 10; 14; 15] adopt integer additions and the LUTs of integer entries for practical consideration. To accelerate the convergence speed, some MIM-FAIDs, i.e., [13; 15; 16], further consider either a layered schedule [22] or a shuffled schedule [23]. For example, the layered MIM-FAID [13] is designed by the IB method for decoding the regular LDPC codes. The LUTs of the FAID in [16] are constructed based on the layered schedule with a high-precision uniform channel quantizer. The MIM quantized shuffled min-sum (MIM-QSMS) decoder was proposed in [15], which designs the LUTs by considering the shuffled decoding schedule. All of these MIM-FAIDs with different decoding schedules, e.g., [15; 16], are designed for a particular LDPC code with a fixed code rate, which cannot be used to decode LDPC codes with different code rates.