  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.
