Sparse superposition codes with rotational invariant coding matrices for memoryless channels

TL;DR

This paper extends the analysis of structured coding matrices for sparse superposition codes to binary memoryless channels, demonstrating their capacity-achieving potential and universal error floor properties using statistical physics methods.

Contribution

It generalizes previous results to binary channels and confirms the spectral criterion's effectiveness for a broad class of memoryless channels.

Findings

Spectral criterion extends to binary memoryless channels.

Structured matrices achieve capacity in large section limit.

Error floor property is universal across matrix spectra.

Abstract

We recently showed in [1] the superiority of certain structured coding matrices ensembles (such as partial row-orthogonal) for sparse superposition codes when compared with purely random matrices with i.i.d. entries, both information-theoretically and under practical vector approximate message-passing decoding. Here we generalize this result to binary input channels under generalized vector approximate message-passing decoding [2].We focus on specific binary output channels for concreteness but our analysis based on the replica symmetric method from statistical physics applies to any memoryless channel. We confirm that the "spectral criterion" introduced in [1], a coding-matrix design principle which allows the code to be capacity-achieving in the "large section size" asymptotic limit, extends to generic memoryless channels. Moreover, we also show that the vanishing error floor property [3] of this coding scheme is universal for arbitrary spectrum of the coding matrix.