  We consider sparse superposition codes (SPARCs) over complex AWGN channels.
Such codes can be efficiently decoded by an approximate message passing (AMP)
decoder, whose performance can be predicted via so-called state evolution in
the large-system limit. In this paper, we mainly focus on how to use
concatenation of SPARCs and cyclic redundancy check (CRC) codes on the encoding
side and use list decoding on the decoding side to improve the finite-length
performance of the AMP decoder for SPARCs over complex AWGN channels.
Simulation results show that such a concatenated coding scheme works much
better than SPARCs with the original AMP decoder and results in a steep
waterfall-like behavior in the bit-error rate performance curves. Furthermore,
we apply our proposed concatenated coding scheme to spatially coupled SPARCs.
Besides that, we also introduce a novel class of design matrices, i.e.,
matrices that describe the encoding process, based on circulant matrices
derived from Frank or from Milewski sequences. This class of design matrices
has comparable encoding and decoding computational complexity as well as very
close performance with the commonly-used class of design matrices based on
discrete Fourier transform (DFT) matrices, but gives us more degrees of freedom
when designing SPARCs for various applications.
