This article introduces a novel concatenated coding scheme called sparse regression LDPC (SR-LDPC) codes. An SR-LDPC code consists of an outer non-binary LDPC code and an inner sparse regression code (SPARC) whose respective field size and section sizes are equal. For such codes, an efficient decoding algorithm is proposed based on approximate message passing (AMP) that dynamically shares soft information between inner and outer decoders. This dynamic exchange of information is facilitated by a denoiser that runs belief propagation (BP) on the factor graph of the outer LDPC code within each AMP iteration. It is shown that this denoiser falls within the class of non-separable pseudo-Lipschitz denoising functions and thus that state evolution holds for the proposed AMP-BP algorithm. Leveraging the rich structure of SR-LDPC codes, this article proposes an efficient low-dimensional approximate state evolution recursion that can be used for efficient hyperparameter tuning, thus paving the way for future work on optimal code design. Finally, numerical simulations demonstrate that SR-LDPC codes outperform contemporary codes over the AWGN channel for parameters of practical interest. SR-LDPC codes are shown to be viable means to obtain shaping gains over the AWGN channel.

翻译：本文提出了一种新颖的级联编码方案——稀疏回归LDPC(SR-LDPC)码。SR-LDPC码由外部非二元LDPC码和内部稀疏回归码(SPARC)构成，两者的域大小与分段大小分别相等。针对此类码字，本文基于近似消息传递(AMP)算法提出了一种高效译码方案，该方案能够在内部与外部译码器之间动态共享软信息。这种动态信息交换通过一个去噪器实现，该去噪器在每次AMP迭代中对外部LDPC码的因子图执行置信传播(BP)算法。研究表明，该去噪器属于非可分离伪Lipschitz去噪函数类，因此所提出的AMP-BP算法满足状态演化特性。借助SR-LDPC码的丰富结构，本文提出了一种高效的低维近似状态演化递推方法，可用于超参数的高效调优，从而为未来最优码设计的研究奠定基础。最后，数值仿真表明，在加性高斯白噪声(AWGN)信道中，对于实际感兴趣的参数，SR-LDPC码的性能优于当代码字。此外，SR-LDPC码被证实是获得AWGN信道成形增益的可行方案。