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Removal of Small Weight Stopping Sets for Asynchronous Unsourced Multiple Access

Frederik Ritter, Jonathan Mandelbaum, Alexander Fengler, Holger Jäkel, Laurent Schmalen

TL;DR

This work studies the error floor in frame-asynchronous two-user unsourced MACs (AU-BAC) caused by small stopping sets in the joint factor graph. It introduces a column-permutation algorithm to completely avoid the smallest stopping sets (4SET) for all delays $\tau \ge 1$, leveraging a necessary condition based on a unique total distance multiset $\mathcal{D}$ and combining with PEG girth optimization for further gains. Numerical results show substantial improvements in PUPE, including almost two orders of magnitude in noiseless channels and up to a factor of $84.2$ at large block lengths when using PEG with degree-one VN stopping-set removal, relative to RCC. The findings highlight the critical role of joint-graph structure in asynchronous unsourced MACs and provide practical design rules to eliminate stopping-set-induced error floors through PCM permutation and structured LDPC constructions.

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.

Removal of Small Weight Stopping Sets for Asynchronous Unsourced Multiple Access

TL;DR

This work studies the error floor in frame-asynchronous two-user unsourced MACs (AU-BAC) caused by small stopping sets in the joint factor graph. It introduces a column-permutation algorithm to completely avoid the smallest stopping sets (4SET) for all delays , leveraging a necessary condition based on a unique total distance multiset and combining with PEG girth optimization for further gains. Numerical results show substantial improvements in PUPE, including almost two orders of magnitude in noiseless channels and up to a factor of at large block lengths when using PEG with degree-one VN stopping-set removal, relative to RCC. The findings highlight the critical role of joint-graph structure in asynchronous unsourced MACs and provide practical design rules to eliminate stopping-set-induced error floors through PCM permutation and structured LDPC constructions.

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.
Paper Structure (11 sections, 3 theorems, 20 equations, 7 figures, 1 table, 2 algorithms)

This paper contains 11 sections, 3 theorems, 20 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Removal of Small Weight Stopping Sets
  4. Properties of 4SET
  5. Refined Removal of 4SET
  6. Numerical Results
  7. Conclusion
  8. Proof of Theorem \ref{['theorem:minimize-d']}
  9. Deterministic Construction for $V\leq2$
  10. Lower Bound on Distance Set Size
  11. Test for Degree-One VN Stopping-Sets

Key Result

Theorem 1

Given a PCM $\bm{H}$, if the corresponding total distance multiset $\mathcal{D}$ contains only unique elements, no 4SET can form in the joint graph for any delay $\tau > 0$. We call this a 4SET-free PCM.

Figures (7)

  • Figure 1: Exemplary joint graph with a delay $\tau = 1$. VN are represented by circles, CN are squares and MN are triangles.
  • Figure 2: A joint graph for $\tau = 2$ with a highlighted 4SET.
  • Figure 3: $(L_1,r_\mathrm{d})$-region with different ensembles and values for $V$ marked.
  • Figure 4: $(L_1,r_\mathrm{d})$-region with bound on points for which a 4SET free PCM exists.
  • Figure 5: PUPE of a $n=500$ code constructed from Ensemble \ref{['ensemble-2']} using PEG for different values of $\tau$. 100000.0 transmissions are simulated per point. The PUPE peak at $\tau = 10$ is caused by a 4SET.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Theorem 1: 4SET-free PCM
  • Theorem 2: Converse
  • Theorem 3