# Generalized Approximate Message-Passing Decoder for Universal Sparse Superposition Codes

## TL;DR

This paper introduces a GAMP decoder for sparse superposition codes, demonstrating their universality across channels, analyzing performance limits, and proposing spatial coupling and Hadamard encoders to approach capacity.

## Contribution

It presents the first GAMP decoding algorithm for SS codes, confirms their universality, and analyzes performance improvements via spatial coupling and practical Hadamard encoders.

## Key findings

- GAMP decoder achieves near-capacity performance with spatial coupling.
- SS codes have a vanishing error floor for binary channels at large code lengths.
- Hadamard-based encoders are effective for practical implementations.

## Abstract

Sparse superposition (SS) codes were originally proposed as a capacity-achieving communication scheme over the additive white Gaussian noise channel (AWGNC) [1]. Very recently, it was discovered that these codes are universal, in the sense that they achieve capacity over any memoryless channel under generalized approximate message-passing (GAMP) decoding [2], although this decoder has never been stated for SS codes. In this contribution we introduce the GAMP decoder for SS codes, we confirm empirically the universality of this communication scheme through its study on various channels and we provide the main analysis tools: state evolution and potential. We also compare the performance of GAMP with the Bayes-optimal MMSE decoder. We empirically illustrate that despite the presence of a phase transition preventing GAMP to reach the optimal performance, spatial coupling allows to boost the performance that eventually tends to capacity in a proper limit. We also prove that, in contrast with the AWGNC case, SS codes for binary input channels have a vanishing error floor in the limit of large codewords. Moreover, the performance of Hadamard-based encoders is assessed for practical implementations.