Removal of Small Weight Stopping Sets for Asynchronous Unsourced Multiple Access

TL;DR

This paper analyzes small stopping sets in asynchronous two-user systems and proposes an algorithm to eliminate them, significantly reducing error floors and improving error performance across delays.

Contribution

It introduces a novel reordering algorithm to avoid small stopping sets in joint factor graphs for asynchronous unsourced multiple access.

Findings

Error floor caused by stopping sets is fully mitigated.

Algorithm improves error probability by nearly two orders of magnitude.

Combining girth optimization with the algorithm yields further improvements.

Abstract

In this paper, we analyze the formation of small stopping sets in joint factor graphs describing a frame-asynchronous two-user transmission. Furthermore, we propose an algorithm to completely avoid small stopping sets in the joint factor graph over the entire range of symbol delays. The error floor caused by these stopping sets is completely mitigated. Our key observation is that, while the order of bits in the codeword is irrelevant in a single-user environment, it turns out to be crucial in an asynchronous, unsourced two-user system. Subsequently, our algorithm finds a reordering of variable nodes which avoids the smallest stopping set in the joint graph. We show that further improvements can be achieved when girth optimization of the single-user graphs by progressive edge growth (PEG) is used in combination with our proposed algorithm. Starting with a randomized code construction with optimized degree distribution, our simulation results show that PEG followed by the proposed algorithm can improve the average per user probability of error in a noiseless channel by almost two orders of magnitude for a broad range of frame delays.