Capacity-achieving Sparse Superposition Codes via Approximate Message Passing Decoding

TL;DR

This paper introduces an approximate message passing decoder for sparse superposition codes, achieving near-capacity communication over AWGN channels with low complexity and improved finite-length performance.

Contribution

The paper presents a linear-complexity decoding algorithm for sparse superposition codes that asymptotically achieves channel capacity, along with a power allocation scheme and Hadamard matrices for enhanced performance.

Findings

Decoder asymptotically achieves AWGN capacity

Power allocation improves empirical performance

Hadamard matrices reduce decoding complexity

Abstract

Sparse superposition codes were recently introduced by Barron and Joseph for reliable communication over the AWGN channel at rates approaching the channel capacity. The codebook is defined in terms of a Gaussian design matrix, and codewords are sparse linear combinations of columns of the matrix. In this paper, we propose an approximate message passing decoder for sparse superposition codes, whose decoding complexity scales linearly with the size of the design matrix. The performance of the decoder is rigorously analyzed and it is shown to asymptotically achieve the AWGN capacity with an appropriate power allocation. Simulation results are provided to demonstrate the performance of the decoder at finite blocklengths. We introduce a power allocation scheme to improve the empirical performance, and demonstrate how the decoding complexity can be significantly reduced by using Hadamard design matrices.