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Variable-Length Feedback Codes over Known and Unknown Channels with Non-vanishing Error Probabilities

Recep Can Yavas, Vincent Y. F. Tan

TL;DR

This work analyzes variable-length feedback codes with noiseless feedback over discrete memoryless channels in the non-vanishing error regime, delivering new non-asymptotic and second-order achievability bounds. It introduces a modified Yamamoto--Itoh scheme with two communication phases and one confirmation phase, augmented by a stop-at-time-zero strategy to achieve the $\epsilon$-capacity while improving the $\log N$-scaled second-order term to $-\frac{C}{C_1}\log N$. The paper also universalizes the VLF construction by replacing the information-density test with an empirical mutual information threshold, yielding a UVLF bound that holds across channel families, aided by refined method-of-types and nonlinear renewal theory arguments. Extensions to Gaussian channels with power constraints are provided, including universal metrics based on empirical correlation that enable reliable operation with unknown noise variance. Collectively, the results tighten fundamental limits for VLF and UVLF coding and offer practically robust universal schemes with improved finite-length performance.

Abstract

We study variable-length feedback (VLF) codes with noiseless feedback for discrete memoryless channels. We present a novel non-asymptotic bound, which analyzes the average error probability and average decoding time of our modified Yamamoto--Itoh scheme. We then optimize the parameters of our code in the asymptotic regime where the average error probability $ε$ remains a constant as the average decoding time $N$ approaches infinity. Our second-order achievability bound is an improvement of Polyanskiy et al.'s (2011) achievability bound. We also universalize our code by employing the empirical mutual information in our decoding metric and derive a second-order achievability bound for universal VLF codes. Our results for both VLF and universal VLF codes are extended to the additive white Gaussian noise channel with an average power constraint. The former yields an improvement over Truong and Tan's (2017) achievability bound. The proof of our results for universal VLF codes uses a refined version of the method of types and an asymptotic expansion from the nonlinear renewal theory literature.

Variable-Length Feedback Codes over Known and Unknown Channels with Non-vanishing Error Probabilities

TL;DR

This work analyzes variable-length feedback codes with noiseless feedback over discrete memoryless channels in the non-vanishing error regime, delivering new non-asymptotic and second-order achievability bounds. It introduces a modified Yamamoto--Itoh scheme with two communication phases and one confirmation phase, augmented by a stop-at-time-zero strategy to achieve the -capacity while improving the -scaled second-order term to . The paper also universalizes the VLF construction by replacing the information-density test with an empirical mutual information threshold, yielding a UVLF bound that holds across channel families, aided by refined method-of-types and nonlinear renewal theory arguments. Extensions to Gaussian channels with power constraints are provided, including universal metrics based on empirical correlation that enable reliable operation with unknown noise variance. Collectively, the results tighten fundamental limits for VLF and UVLF coding and offer practically robust universal schemes with improved finite-length performance.

Abstract

We study variable-length feedback (VLF) codes with noiseless feedback for discrete memoryless channels. We present a novel non-asymptotic bound, which analyzes the average error probability and average decoding time of our modified Yamamoto--Itoh scheme. We then optimize the parameters of our code in the asymptotic regime where the average error probability remains a constant as the average decoding time approaches infinity. Our second-order achievability bound is an improvement of Polyanskiy et al.'s (2011) achievability bound. We also universalize our code by employing the empirical mutual information in our decoding metric and derive a second-order achievability bound for universal VLF codes. Our results for both VLF and universal VLF codes are extended to the additive white Gaussian noise channel with an average power constraint. The former yields an improvement over Truong and Tan's (2017) achievability bound. The proof of our results for universal VLF codes uses a refined version of the method of types and an asymptotic expansion from the nonlinear renewal theory literature.
Paper Structure (23 sections, 10 theorems, 89 equations, 3 figures, 1 table)

This paper contains 23 sections, 10 theorems, 89 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Main Contributions
  3. Paper Organization
  4. Notation and Definitions
  5. Problem Formulation
  6. Main Result
  7. Extension to the Gaussian Channel
  8. Proofs of Theorems \ref{['thm:nonasymp']} and \ref{['thm:VLF']}
  9. Proof of Theorem \ref{['thm:nonasymp']}
  10. Coding scheme
  11. Error probability analysis
  12. Average decoding time analysis
  13. Proof of Theorem \ref{['thm:VLF']}
  14. Proof of Theorem \ref{['thm:UVLF']}
  15. Supporting Lemmas
  16. ...and 8 more sections

Key Result

Theorem 1

Let $P_{Y|X}$ be the underlying DMC with $C_1 < \infty$ and $C > 0$. Fix a positive integer $M$, positive constants $\gamma_1 < \gamma_2$, $a_{\mathrm{A}}$, and $a_{\mathrm{R}}$, $\epsilon_0 \in (0, 1)$, and a capacity-achieving input distribution $P_X$. Define There exists an $(N, M, \epsilon)$-VLF code with where $(X, Y) \sim P_X P_{Y|X}$, $Y_{\mathrm{A}} \sim P_{Y|X = x_{\mathrm{A}}}$, $Y_{\

Figures (3)

  • Figure 1: Achievable rates over the cascade of a BSC with flip probability 0.11 and a BEC with erasure probability 0.2 are shown. The target error probability is $\epsilon = 10^{-3}$ for both figures. The average decoding time $N$ ranges in [100, 1500] for (a) and in $[5 \times 10^4, 2.5 \times 10^5]$ for (b).
  • Figure 2: Achievable rates over the BSC with flip probability 0.11 are shown. The target error probability is $\epsilon = 10^{-3}$, and the average decoding time is $N \in [100, 1500]$.
  • Figure 3: Asymptotic expansions in Theorems \ref{['thm:VLF']}--\ref{['thm:UVLF']} (a) over the cascade of BSC(0.11) and BEC(0.2) and (b) over BSC(0.11) are shown. The average decoding time is $N \in [100, 1500]$, and the error probability is $\epsilon = 10^{-3}$.

Theorems & Definitions (13)

  • Definition 1: VLF code polyanskiy2011feedback
  • Definition 2: UVLF code
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 3
  • Theorem 4
  • Theorem 5
  • Lemma 1: lorden1970, gutbook
  • Lemma 2
  • ...and 3 more