Efficient decoding is crucial to high-throughput and low-power wireless communication scenarios. A theoretical analysis of the performance-complexity tradeoff toward low-complexity decoding is required for a better understanding of the fundamental limits in the above-mentioned scenarios. This study aims to explore the performance of decoders with complexity constraints. Specifically, we investigate the performance of LDPC codes with different numbers of belief-propagation iterations and the performance of polar codes with an SSC decoder. We found that the asymptotic error rates of both polar codes and LDPC codes are functions of complexity $T$ and code length $N$, in the form of $2^{-a2^{b\frac{T}{N}}}$, where $a$ and $b$ are constants that depend on channel and coding schemes. Our analysis reveals the different performance-complexity tradeoffs for LDPC and polar codes. The results indicate that if one aims to further enhance the decoding efficiency for LDPC codes, the key lies in how to efficiently pass messages on the factor graph. In terms of decoding efficiency, polar codes asymptotically outperform $(J, K)$-regular LDPC codes with a code rate $R \le 1-\frac{J(J-1)}{2^J+(J-1)}$ in the low-complexity regime $(T \le O(NlogN))$.

翻译：高效译码对于高吞吐量、低功耗无线通信场景至关重要。为深入理解上述场景中的基本极限，需对低复杂度译码的性能-复杂度权衡进行理论分析。本研究旨在探索复杂度约束下译码器的性能表现。具体而言，我们考察了采用不同置信传播迭代次数的LDPC码性能，以及使用SSC译码器的Polar码性能。研究发现，Polar码与LDPC码的渐近误码率均为复杂度$T$与码长$N$的函数，其形式可表示为$2^{-a2^{b\frac{T}{N}}}$，其中$a$与$b$为依赖于信道与编码方案的常数。分析揭示了LDPC码与Polar码在性能-复杂度权衡上的差异。结果表明，若要进一步提升LDPC码的译码效率，关键在于如何在因子图上高效传递信息。就译码效率而言，在低复杂度区域$(T \le O(NlogN))$内，Polar码在码率$R \le 1-\frac{J(J-1)}{2^J+(J-1)}$条件下渐近优于$(J, K)$正则LDPC码。