  Belief propagation applied to iterative decoding and sparse recovery through
approximate message passing (AMP) are two research areas that have seen
monumental progress in recent decades. Inspired by these advances, this article
introduces sparse regression LDPC codes and their decoding. Sparse regression
codes (SPARCs) are a class of error correcting codes that build on ideas from
compressed sensing and can be decoded using AMP. In certain settings, SPARCs
are known to achieve capacity; yet, their performance suffers at finite block
lengths. Likewise, LDPC codes can be decoded efficiently using belief
propagation and can also be capacity achieving. This article introduces a novel
concatenated coding structure that combines an LDPC outer code with a
SPARC-inspired inner code. Efficient decoding for such a code can be achieved
using AMP with a denoiser that performs belief propagation on the factor graph
of the outer LDPC code. The proposed framework exhibits performance
improvements over SPARCs and standard LDPC codes for finite block lengths and
results in a steep waterfall in error performance, a phenomenon not observed in
uncoded SPARCs. Findings are supported by numerical results.
