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IEEE Transactions on Communications

An Analysis and Design of Rate-Dependent Nested Scheduling in Layered Decoding of LDPC Codes

Dongxu Chang, Peng Liu, Guanghui Wang, Guiying Yan, Zhi-Ming Ma

Published Jul 3, 2025• 2 citations
Publisher

Abstract

In this study, we analyze the characteristics of scheduling sequences for layered belief propagation (LBP) that can result in efficient decoding of low-density parity-check (LDPC) codes. Specifically, we claim that scheduling sequences leading to high decoding efficiency should prioritize updating check nodes with lower error probabilities aggregated from neighboring variable nodes. We prove this conclusion separately on both the BEC and the BI-AWGN channels. Some observable characteristics in “good” scheduling sequences regarding row weights, rows connected to punctured columns, and column weights can serve as corollaries to this conclusion. By comprehensively considering these characteristics of good scheduling, we design a multi-sequence nested scheduling scheme of layered decoding for 5G New Radio (NR) LDPC codes. The proposed schemes can obtain scheduling sequences for various rates of rate-compatible LDPC codes using small hardware storage. What’s more, by respectively storing double-sequence, triple-sequence, or more sequences to obtain scheduling sequences at various code rates, a trade-off can be made between decoder storage and decoding performance. Experimental results demonstrate that the proposed scheme achieves performance improvements compared to existing scheduling schemes at nearly all code rates.

DOI
10.1109/tcomm.2025.3585512

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In this study, we analyze the characteristics of scheduling sequences for layered belief propagation (LBP) that can result in efficient decoding of low-density parity-check (LDPC) codes. Specifically, we claim that scheduling sequences leading to high decoding efficiency should prioritize updating check nodes with lower error probabilities aggregated from neighboring variable nodes. We prove this conclusion separately on both the BEC and the BI-AWGN channels. Some observable characteristics in “good” scheduling sequences regarding row weights, rows connected to punctured columns, and column weights can serve as corollaries to this conclusion. By comprehensively considering these characteristics of good scheduling, we design a multi-sequence nested scheduling scheme of layered decoding for 5G New Radio (NR) LDPC codes. The proposed schemes can obtain scheduling sequences for various rates of rate-compatible LDPC codes using small hardware storage. What’s more, by respectively storing double-sequence, triple-sequence, or more sequences to obtain scheduling sequences at various code rates, a trade-off can be made between decoder storage and decoding performance. Experimental results demonstrate that the proposed scheme achieves performance improvements compared to existing scheduling schemes at nearly all code rates.
In this study, we analyze the characteristics of scheduling sequences for Layered belief propagation (LBP) that can result in efficient decoding on Low-density parity-check (LDPC) codes. Specifically, we claim that scheduling sequences leading to high decoding efficiency should prioritize updating check nodes with lower error probabilities aggregated from neighboring variable nodes. We prove this conclusion separately in both BEC and BI-AWGN channels. Some observable characteristics in "good" scheduling sequences regarding row weights, rows connected to punctured columns, and column weights can serve as corollaries to this conclusion. Based on these, we design a dynamic scheduling algorithm, which can further demonstrate the effectiveness of these characteristics through simulation experiments.
Low-density parity-check (LDPC) codes have been chosen for the fifth-generation (5G) new radio (NR) enhanced Mobile Broad Band (eMBB) data channel. Since NR LDPC codes puncture the first two block columns of the parity-check matrix, the decoding complexity is reduced, but it slows down the convergence rate. In this letter, we propose a fixed schedule that decodes the layers with the least-punctured edges and those with the highest-degree (LPHD), which has much faster convergence speed than conventional schemes. The fixed scheduling schemes cannot take full advantage of the dynamic changes of decoding messages. As a result, the residual-based layered belief propagation (RB-LBP) is proposed to rearrange the layers dynamically among different iterations. Compared with some informed dynamic scheduling (IDS) strategies, RB-LBP reduces the decoding complexity with slight performance loss.
Schedule plays an important role in layered belief propagation (BP) decoding of low-density parity-check (LDPC) codes. It affects the error-rate performance as well as convergence rate of the decoder. In this paper, an efficient scheduling scheme is proposed to arrange the updating order of check nodes for layered BP decoder of regular LDPC codes. This scheme selects check node one at a time based on the reliability of the variable nodes. Specifically each time the check node with maximum number of low-reliability neighbors is selected. For simplicity, we introduce the updating state instead of the real-time messages to represent variable-node reliability. Simulation results on regular LDPC codes prove that the proposed scheduling scheme can obtain efficient schedules with fast convergence and good error-rate performance.
An efficient dynamic schedule for layered belief-propagation (LBP) decoding of low-density parity-check (LDPC) codes is presented, in which the check nodes connecting to a variable node with maximum relative message residual are chosen in each dynamic decoding iteration, and then LBP is performed for these check nodes. In this schedule we combined the features of lazy schedule and node-wise residual belief propagation (NWRBP) together while keeping the extra computational complexity very low. Simulation results show that the new schedule speeds up the convergence rate and greatly improves error performance at medium to high signal-to-noise ratio (SNR) when compared to the standard LBP decoding algorithm.
Low-density parity-check (LDPC) codes have been selected as the channel coding scheme for 5G NR standard, which have excellent performance approaching the Shannon limit and are easy to implement in hardware. The decoding algorithm and scheduling scheme of LDPC codes have a significant impact on performance. The row degree (RD) and the number of punctured bits (PBs) are widely used to determine the update order of layers. In this letter, we used Matlab to simulate the error performance of the 5G LDPC codes. We completed the bit error rate and frame error rate simulations of six different decoding algorithms, including flooding LLR-BP, flooding NMS, layered LLR-BP, layered NMS, the lowest degree and least-punctured (LD-LP) layered LLR-BP and LD-LP layered NMS methods, for the 5G LDPC code with a base graph of size [46, 68], code rate 1/3, and code length of 1584.We further compared the performance of LD-LP layered decoding and LP-LD layered decoding. Moreover, we simulated the error performance of the core part of the LDPC code, which consists of four layers. We evaluated the effect of the update order of the four layers on the error performance of the code, and concluded that some update orders might give slightly better error performance than others.
In this study, a scheduling policy of layered decoding for quasi-cycle (QC) low-density parity-check (LDPC) codes with high throughput and good performance is designed. The influence of scheduling on the delay of the decoder's hardware implementation and on the decoding performance are considered simultaneously. Specifically, we analyze the idle time required under various scheduling sequences within a pipelined decoding architecture and formulate the problem as a traveling salesman problem (TSP) aiming at minimizing idle time. Furthermore, considering that different scheduling sequences can affect decoding performance, we refine the graph used to solve the TSP based on scheduling characteristics that promote improved decoding outcomes. Simulation results demonstrate that the identified scheduling sequence achieves a low number of hardware delays while maintaining excellent decoding performance for 5G New Radio (NR) LDPC codes.