Decoding of Low-Density Parity Check (LDPC) codes can be viewed as a special case of XOR-SAT problems, for which low-computational complexity bit-flipping algorithms have been proposed in the literature. However, a performance gap exists between the bit-flipping LDPC decoding algorithms and the benchmark LDPC decoding algorithms, such as the Sum-Product Algorithm (SPA). In this paper, we propose an XOR-SAT solver using log-sum-exponential functions and demonstrate its advantages for LDPC decoding. This is then approximated using the Margin Propagation formulation to attain a low-complexity LDPC decoder. The proposed algorithm uses soft information to decide the bit-flips that maximize the number of parity check constraints satisfied over an optimization function. The proposed solver can achieve results that are within $0.1$dB of the Sum-Product Algorithm for the same number of code iterations. It is also at least 10x lesser than other Gradient-Descent Bit Flipping decoding algorithms, which are also bit-flipping algorithms based on optimization functions. The approximation using the Margin Propagation formulation does not require any multipliers, resulting in significantly lower computational complexity than other soft-decision Bit-Flipping LDPC decoders.

翻译：低密度奇偶校验码的解码可视为异或可满足性问题的特例，现有文献已提出低计算复杂度的比特翻转算法。然而，比特翻转型LDPC解码算法与标杆算法（如和积算法）之间仍存在性能差距。本文提出一种采用对数-求和-指数函数的异或可满足性求解器，并证明其在LDPC解码中的优势。通过边际传播公式进行近似处理后，该求解器可实现低复杂度的LDPC解码器。所提算法利用软信息决定比特翻转操作，以最大化优化函数中满足的奇偶校验约束数量。在相同码字迭代次数下，该求解器与和积算法的性能差距在0.1dB以内，且其复杂度至少比同为基于优化函数的梯度下降比特翻转解码算法低一个数量级。基于边际传播公式的近似方案无需任何乘法器，因此相比其他软判决比特翻转LDPC解码器具有显著更低的计算复杂度。