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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 inthe large-system limit. In this paper, we mainly focus on how to useconcatenation of SPARCs and cyclic redundancy check (CRC) codes on the encodingside and use list decoding on the decoding side to improve the finite-lengthperformance of the AMP decoder for SPARCs over complex AWGN channels.Simulation results show that such a concatenated coding scheme works muchbetter than SPARCs with the original AMP decoder and results in a steepwaterfall-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 matricesderived from Frank or from Milewski sequences. This class of design matriceshas comparable encoding and decoding computational complexity as well as veryclose performance with the commonly-used class of design matrices based ondiscrete Fourier transform (DFT) matrices, but gives us more degrees of freedomwhen designing SPARCs for various applications.

Using List Decoding to Improve the Finite-Length Performance of Sparse Regression Codes