## 1 Introduction  
Satellite communication technology is a communication system that uses artificial satellites as the relay and inter-satellite and satellite-to-ground links as transmission networks to obtain, transmit, and process space information [1]. At the same time, satellite communications have excellent coverage and communication capacity. Whether it is land, sea, or air, it can realize communication connections and fill the areas that cannot be covered by ground infrastructure. It is often used for disaster-relief emergency communication and the real-time transmission of battlefield information [2]. Due to the long distance of satellite communications, the signal will be significantly attenuated during transmission, and the communication link will be affected by weather conditions such as clouds and rain [3], resulting in a high bit error rate (BER). Channel coding technology, as a key underlying technology in communication systems, can improve anti-interference and reliability [4]. The essence of channel coding technology is to introduce some controllable redundancy in the transmitted information, and the receiver uses this redundancy to detect and recover the transmission error so as to ensure efficient and reliable data transmission. In recent years, achieving high-speed and reliable transmission with limited hardware resources on satellite has become a core challenge in the development of satellite communication. Different encoding and decoding techniques can significantly reduce the signal-to-noise ratio (SNR) requirement under the same BER [5] (\(SNR=10\lg(P_{s}/P_{n})\)where \(P_{s}\) and \(P_{n}\) represent the effective power of the signal and noise, respectively), thereby reducing the transmission power of satellite communication equipment and making it possible to miniaturize. For example, in satellite navigation, every 1 dB of SNR gain brought about by channel coding is converted into profits of billions of dollars [6]. Low-density parity-check (LDPC) codes have the performance of approaching the Shannon limit [7], high coding gain, and strong anti-interference ability [8]. They have been incorporated into the channel coding standard by the consultative committee for space data systems (CCSDS). Compared with traditional RS codes and convolutional codes, LDPC codes have the advantages of a simple coding structure, easy hardware implementation, and a low decoding threshold [9]. Compared with Turbo codes, LDPC codes have low coding complexity and are more suitable for parallel high-throughput encoder-decoder design [10]. LDPC codes have flexible bit-rate structures [11] and low error levels [12] and meet the application scenario of satellite link communication, so they have been widely used in the field of satellite communications [13]. For instance, reference [14] presented a novel early termination stopping criterion for LDPC decoders that is based on detecting the periodicity of syndrome weight oscillations. This design can reduce computational energy and can be implemented to meet the energy resources of satellite communications. Reference [15] presented filtered orthogonal frequency division multiplexed LDPC codes, which can achieve a coding gain of 1.2 dB with a BER of \(10^{-3}\) under moderate rain weather conditions and can be well applied to Ka-band satellite communications. In reference [16], a decoding algorithm based on majority logic for non-binary low-density parity-check (NIB-LDPC) codes was presented, including Algorithm B (AlgB) and weighted-Algorithm B (wtd-AlgB). The AlgB decoding algorithm only considers the extrinsic information frequency, and its performance is not as good as the performance of the wtd-AlgB decoding algorithm. The wtd-AlgB decoding algorithm combines the Hamming distance coefficient and the extrinsic information to form a reliable measure for hard-decision symbols. However, the algorithm requires frequent multiplication operations for each decision.