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Undetected Error Probability in the Short Blocklength Regime: Approaching Finite-Blocklength Bounds with Polar Codes

Alexander Sauter, A. Oguz Kislal, Giuseppe Durisi, Gianluigi Liva, Balazs Matuz, Erik G. Ström

TL;DR

This work tackles undetected errors in short-packet communications by deriving two finite-blocklength achievability bounds for CRC-aided polar codes: one using an outer CRC for error detection and another based on thresholding the generalized information density (Forney-like rule). The authors show that, on biAWGN channels, the threshold-based approach yields tighter bounds and improved performance for short blocklengths, while the CRC-based method offers robustness under mismatched channel-state information. They also provide saddlepoint approximations to efficiently compute tail probabilities, enabling practical benchmarking of TEP and UEP in URLLC scenarios. The results offer actionable design guidance: threshold-based detection is preferable in well-characterized channels, whereas CRC-based detection is more reliable when CSI is imperfect. Overall, the paper advances finite-blocklength theory and its application to practical, reliable short-packet coding with polar codes.

Abstract

We analyze the trade-off between the undetected error probability (i.e., the probability that the channel decoder outputs an erroneous message without detecting the error) and the total error probability in the short blocklength regime. We address the problem by developing two new finite blocklength achievability bounds, which we use to benchmark the performance of two coding schemes based on polar codes with outer cyclic redundancy check (CRC) codes -- also referred to as CRC-aided (CA) polar codes. The first bound is obtained by considering an outer detection code, whereas the second bound relies on a threshold test applied to the generalized information density. Similarly, in the first CA polar code scheme, we reserve a fraction of the outer CRC parity bits for error detection, whereas in the second scheme, we apply a threshold test (specifically, Forney's optimal rule) to the output of the successive cancellation list decoder. Numerical simulations performed on the binary-input AWGN channel reveal that, in the short-blocklength regime, the threshold-based approach is superior to the CRC-based approach, both in terms of bounds and performance of CA polar code schemes. We also consider the case of decoding with noisy channel-state information, which leads to a mismatched decoding setting. Our results illustrate that, differently from the previous case, in this scenario, the CRC-based approach outperforms the threshold-based approach, which is more sensitive to the mismatch.

Undetected Error Probability in the Short Blocklength Regime: Approaching Finite-Blocklength Bounds with Polar Codes

TL;DR

This work tackles undetected errors in short-packet communications by deriving two finite-blocklength achievability bounds for CRC-aided polar codes: one using an outer CRC for error detection and another based on thresholding the generalized information density (Forney-like rule). The authors show that, on biAWGN channels, the threshold-based approach yields tighter bounds and improved performance for short blocklengths, while the CRC-based method offers robustness under mismatched channel-state information. They also provide saddlepoint approximations to efficiently compute tail probabilities, enabling practical benchmarking of TEP and UEP in URLLC scenarios. The results offer actionable design guidance: threshold-based detection is preferable in well-characterized channels, whereas CRC-based detection is more reliable when CSI is imperfect. Overall, the paper advances finite-blocklength theory and its application to practical, reliable short-packet coding with polar codes.

Abstract

We analyze the trade-off between the undetected error probability (i.e., the probability that the channel decoder outputs an erroneous message without detecting the error) and the total error probability in the short blocklength regime. We address the problem by developing two new finite blocklength achievability bounds, which we use to benchmark the performance of two coding schemes based on polar codes with outer cyclic redundancy check (CRC) codes -- also referred to as CRC-aided (CA) polar codes. The first bound is obtained by considering an outer detection code, whereas the second bound relies on a threshold test applied to the generalized information density. Similarly, in the first CA polar code scheme, we reserve a fraction of the outer CRC parity bits for error detection, whereas in the second scheme, we apply a threshold test (specifically, Forney's optimal rule) to the output of the successive cancellation list decoder. Numerical simulations performed on the binary-input AWGN channel reveal that, in the short-blocklength regime, the threshold-based approach is superior to the CRC-based approach, both in terms of bounds and performance of CA polar code schemes. We also consider the case of decoding with noisy channel-state information, which leads to a mismatched decoding setting. Our results illustrate that, differently from the previous case, in this scenario, the CRC-based approach outperforms the threshold-based approach, which is more sensitive to the mismatch.

Paper Structure

This paper contains 19 sections, 3 theorems, 31 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. System Model
  5. Non-asymptotic Achievability Bounds on TEP and UEP
  6. Optimum Erasure Decoder
  7. Achievability Bound via CRC Outer Code
  8. Achievability Bound via Generalized Information Density Thresholding
  9. Saddlepoint Approximation of Pairwise Error Probability
  10. Error Detection for CA Polar Codes
  11. CA Polar Codes: SCL Decoding and Error Detection
  12. SCL Decoding with Split CRC (Algorithm A)
  13. SCL Decoding with Threshold Test (Algorithm B)
  14. Numerical Results
  15. Results for the Binary-Input AWGN Channel
  16. ...and 4 more sections

Key Result

Theorem 1

For an arbitrary input distribution $P_\mathbf{x}\xspace\xspace$ and for every $\Delta \in \mathbb{N}_0$, there exists an $(n,k,\epsilon_{\scaleto{\mathsf{T}}{3.5pt}},\epsilon_{\scaleto{\mathsf{U}}{3.5pt}})$-code for the channel $P_{\mathbf{y}\xspace\xspace|\mathbf{x}\xspace\xspace}(\mathbf{y}\xspac with and

Figures (9)

  • Figure 1: Reference model describing the encoding with a CA polar code, transmission over the channel, and SCL decoding.
  • Figure 2: Decomposition of the outer CRC code $\mathcal{C}_{\scaleto{\mathsf{O}}{3.5pt}}$ as the concatenation of two binary linear block codes.
  • Figure 3: The minimum ${E_b}/{N_0}$ to achieve $\epsilon_{\scaleto{\mathsf{T}}{3.5pt}}^{\star}=10^{-3}$ and $\epsilon_{\scaleto{\mathsf{U}}{3.5pt}}^{\star}=10^{-5}$ as a function of blocklength $n$. Here, we consider a biAWGN channel with $R=0.5$ bits per channel use.
  • Figure 4: TEP and UEP versus ${E_b}/{N_0}$ for $(128, 64)$ for CA polar codes under Algorithm A and Algorithm B ($L=32$). The bounds of Theorem 1 and Theorem 2 are included as a reference. Solid lines are used for the TEP, while dashed lines are used for the UEP. The target TEP and UEP used for the optimization of the error detection parameters are $\epsilon_{\scaleto{\mathsf{T}}{3.5pt}}^* = 10^{-3}$ and $\epsilon_{\scaleto{\mathsf{T}}{3.5pt}}^* = 10^{-5}$.
  • Figure 5: The minimum ${E_b}/{N_0}$ to achieve $\epsilon_{\scaleto{\mathsf{T}}{3.5pt}}^{\star}=10^{-3}$ and $\epsilon_{\scaleto{\mathsf{U}}{3.5pt}}^{\star}=10^{-5}$ as a function of blocklength $n$. The CRC polynomials and the frozen bit set are chosen according to 3GPP21. A CRC-$6$ is used for $n=64$, while a CRC-$11$ is used for $n=128,256$. Here, we consider a biAWGN channel with $R=0.5$ bits per channel use.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 3
  • Remark 5
  • Remark 6
  • ...and 2 more