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Confusions and Erasures of Error-Bounded Block Decoders with Finite Blocklength

Bin Han, Yao Zhu, Rafael F. Schaefer, Giuseppe Caire, Anke Schmeink, H. Vincent Poor, Hans D. Schotten

TL;DR

The paper distinguishes between undetected block confusions and block erasures in AWGN channels under finite blocklength, using a sphere-packing framework to bound both phenomena under error-bounded ML decoding. It derives explicit lower and upper bounds on the block-confusion rate in terms of minimum distances and pairwise confusion probabilities, and analyzes the sensitivity of these bounds to blocklength and power. The key finding is that block confusions are negligibly small compared with block erasures for practical FBL codes, especially at large n and high SNR, thereby supporting the common cross-layer abstraction of treating PHY errors as erasures. This provides a theoretical justification for erasure-based MAC and network-layer models and informs code design in URLLC contexts, while suggesting extensions to broader codebook families and nonuniform distributions.

Abstract

This paper investigates two distinct types of block errors - undetected errors (confusions) and erasures - in additive white Gaussian noise (AWGN) channels with error-bounded block decoders operating in the finite blocklength (FBL) regime. While block error rate (BLER) is a common metric, it does not distinguish between confusions and erasures, which can have significantly different impacts in cross-layer protocol design, despite upper-layer protocols universally assuming physical (PHY) errors manifest as packet erasures rather than undetected corruptions - an assumption lacking rigorous PHY-layer validation. We present a systematic analysis of confusions and erasures under BLER-constrained maximum likelihood (ML) decoding. Through sphere-packing analysis, we provide analytical bounds for both block confusion and erasure probabilities, and derive the sensitivities of these bounds to blocklength and signal-to-noise ratio (SNR). To the best of our knowledge, this is the first study on this topic in the FBL regime. Our findings provide theoretical validation for the block erasure channel abstraction commonly assumed in medium access control (MAC) and network layer protocols, confirming that, for practical FBL codes, block confusions are negligible compared to block erasures, especially at large blocklengths and high SNR.

Confusions and Erasures of Error-Bounded Block Decoders with Finite Blocklength

TL;DR

The paper distinguishes between undetected block confusions and block erasures in AWGN channels under finite blocklength, using a sphere-packing framework to bound both phenomena under error-bounded ML decoding. It derives explicit lower and upper bounds on the block-confusion rate in terms of minimum distances and pairwise confusion probabilities, and analyzes the sensitivity of these bounds to blocklength and power. The key finding is that block confusions are negligibly small compared with block erasures for practical FBL codes, especially at large n and high SNR, thereby supporting the common cross-layer abstraction of treating PHY errors as erasures. This provides a theoretical justification for erasure-based MAC and network-layer models and informs code design in URLLC contexts, while suggesting extensions to broader codebook families and nonuniform distributions.

Abstract

This paper investigates two distinct types of block errors - undetected errors (confusions) and erasures - in additive white Gaussian noise (AWGN) channels with error-bounded block decoders operating in the finite blocklength (FBL) regime. While block error rate (BLER) is a common metric, it does not distinguish between confusions and erasures, which can have significantly different impacts in cross-layer protocol design, despite upper-layer protocols universally assuming physical (PHY) errors manifest as packet erasures rather than undetected corruptions - an assumption lacking rigorous PHY-layer validation. We present a systematic analysis of confusions and erasures under BLER-constrained maximum likelihood (ML) decoding. Through sphere-packing analysis, we provide analytical bounds for both block confusion and erasure probabilities, and derive the sensitivities of these bounds to blocklength and signal-to-noise ratio (SNR). To the best of our knowledge, this is the first study on this topic in the FBL regime. Our findings provide theoretical validation for the block erasure channel abstraction commonly assumed in medium access control (MAC) and network layer protocols, confirming that, for practical FBL codes, block confusions are negligible compared to block erasures, especially at large blocklengths and high SNR.

Paper Structure

This paper contains 25 sections, 15 theorems, 55 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Works
  3. Error-Bounded Decoding with Erasures
  4. Bound Analysis
  5. Bounds of the Minimum Distance between Codewords
  6. Lower Bound of the Block Confusion Rate
  7. Upper Bound of the Block Confusion Rate
  8. Sensitivity of the Confusion Rate Bounds
  9. Lower Bound versus Power
  10. Lower Bound versus Blocklength
  11. Upper Bound versus Power
  12. Upper Bound versus Blocklength
  13. Numerical Results
  14. Conclusion and Outlooks
  15. Proof of Lemma \ref{['lem:dmin_bounds']}
  16. ...and 10 more sections

Key Result

Theorem 1

Given fixed $\mathcal{X}$ with certain $(\mathcal{X}, E,\sigma^2)$, both $\varepsilon$ and $P_{\mathrm{ers}}$ are monotonically decreasing w.r.t. $R$, while $P_{\mathrm{con}}$ monotonically increases.

Figures (5)

  • Figure 1: Schematic representation of typical decision regions: (a) ordinary decoding, (b) erasure option, and (c) list option.
  • Figure 2: fbl codewords projected onto a hyper-sphere
  • Figure 3: Decision spheres of two distinct codewords
  • Figure 4: Block confusion rate bounds vs. $n$ with $E_{\mathrm{b}}/N_0=0\dB$.
  • Figure 5: Block confusion rate bounds vs. $E_{\mathrm{b}}/N_0$. Note the $P_{\mathrm{con}}^{\mathrm{LB}}$ is convex as proven in Theorem \ref{['th:monotonicity_Pcon_LB_regarding_E']}, but may look concave here due to the logarithmic $y$-scale.

Theorems & Definitions (40)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 30 more