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Error Floor of ML-Decoded Spinal Codes in the Finite Blocklength Regime

Aimin Li, Shaohua Wu, Xiaomeng Chen, Sumei Sun

TL;DR

Spinal codes exhibit an error floor in the finite-blocklength regime due to collisions arising from their random coding structure. The authors derive an explicit error-floor expression $P_{EF}$ and a lower bound on $F(L_a,\sigma)$, showing the floor persists as $\sigma^2 \to 0$, and they establish an SNR threshold $\gamma^{th}$ with closed-form forms for Nakagami-$m$, Rayleigh, and AWGN channels. The threshold depends on the modulation order $c$ and channel type but is invariant to the number of passes $L$, while increasing $L$ lowers the floor. These results inform practical power allocation and operating regimes for Spinal codes and motivate collision-mitigation strategies to preserve rateless operation under finite blocklengths.

Abstract

Spinal codes is a new family of capacity-achieving rateless codes that has been shown to achieve better rate performance compared to Raptor codes, Strider codes, and rateless Low-Density Parity-Check (LDPC) codes. This correspondence addresses the performance limitations of Spinal codes in the finite block length regime, uncovering an error floor phenomenon at high Signal-to-Noise Ratios (SNRs). We develop an analytical expression to approximate the error floor and devise SNR thresholds at which the error floor initiates. Numerical results across {Additive White Gaussian Noise (AWGN), rayleigh, and nakagami-m fading channels} verify the accuracy of our analysis. The analysis and numerical results also show that transmitting more passes of symbols can lower the error floor but does not affect the SNR threshold, providing insights on the performance target, the working SNR region, and the code design.

Error Floor of ML-Decoded Spinal Codes in the Finite Blocklength Regime

TL;DR

Spinal codes exhibit an error floor in the finite-blocklength regime due to collisions arising from their random coding structure. The authors derive an explicit error-floor expression and a lower bound on , showing the floor persists as , and they establish an SNR threshold with closed-form forms for Nakagami-, Rayleigh, and AWGN channels. The threshold depends on the modulation order and channel type but is invariant to the number of passes , while increasing lowers the floor. These results inform practical power allocation and operating regimes for Spinal codes and motivate collision-mitigation strategies to preserve rateless operation under finite blocklengths.

Abstract

Spinal codes is a new family of capacity-achieving rateless codes that has been shown to achieve better rate performance compared to Raptor codes, Strider codes, and rateless Low-Density Parity-Check (LDPC) codes. This correspondence addresses the performance limitations of Spinal codes in the finite block length regime, uncovering an error floor phenomenon at high Signal-to-Noise Ratios (SNRs). We develop an analytical expression to approximate the error floor and devise SNR thresholds at which the error floor initiates. Numerical results across {Additive White Gaussian Noise (AWGN), rayleigh, and nakagami-m fading channels} verify the accuracy of our analysis. The analysis and numerical results also show that transmitting more passes of symbols can lower the error floor but does not affect the SNR threshold, providing insights on the performance target, the working SNR region, and the code design.
Paper Structure (14 sections, 5 theorems, 29 equations, 4 figures, 1 table)

This paper contains 14 sections, 5 theorems, 29 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries of Spinal Codes
  3. Encoding of Spinal Codes
  4. ML Decoding of Spinal Codes
  5. Error Floor and SNR Threshold Analysis
  6. Error Floor Analysis of Spinal Codes
  7. SNR Threshold of the Error Floor
  8. Numerical Verification
  9. Error Floor
  10. SNR Threshold
  11. Conclusion
  12. Proof of Lemma \ref{['l1']}
  13. Proof of Theorem \ref{['the3']}
  14. Proof of Corollary \ref{['co1']}

Key Result

Lemma 1

(Restatement of li2024newchen2023tight) Consider $(n,k,c,\Psi,L)$For short-hand notations, we call Spinal codes with message length $n$, segmentation length $k$, modulation order $c$, channel input set $\Psi$, and transmitted passes $L$ as $(n,k,c,\Psi,L)$ Spinal codes. Spinal codes transmitted over with $\mathscr{F} \left(L_a , \sigma \right)$ equals to where $b_t = \frac{\theta_t-\theta_{t-1}}{

Figures (4)

  • Figure 1: Encoding process of Spinal codes.
  • Figure 2: Examples of the constellation mapping set $\Psi$. The left panel is plotted under QAM modulation with $c=6$; The right panel is plotted under QAM modulation with $c=8$.
  • Figure 3: BLER curves of Spinal codes where $n=32$ and $k=4$.
  • Figure 4: BLER curves of Spinal codes where $n=256$, $k=32$, and $c=6$.

Theorems & Definitions (12)

  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • proof
  • ...and 2 more