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Extension of the Poltyrev Bound to Binary Memoryless Symmetric Channels

Tal Philosof, Ariel Doubchak, Amit Berman, Uri Erez

TL;DR

The paper extends the Poltyrev bound, a tight upper bound on decoding error probability, from its original BSC/AWGN settings to general binary memoryless symmetric channels with discrete outputs by employing the method of types. It derives a general, type-based bound that can be evaluated with polynomial-time complexity in the block length and demonstrates it on a hybrid BSC-BEC channel; a reduced-complexity version with linear-time cost is also presented. The approach yields explicit, channel-class-specific bounds for BSC, BEC, hybrid BSC-BEC, and a quinary channel, and shows significant improvements over existing bounds (e.g., Shulman-Feder, Miller-Burshtein) for moderate-length codes such as BCH codes. The results are particularly relevant for systems with small output alphabets and coarse quantization, such as certain flash-memory read channels, where accurate performance metrics are essential for code design and evaluation.

Abstract

The Poltyrev bound provides a very tight upper bound on the decoding error probability when using binary linear codes for transmission over the binary symmetric channel and the additive white Gaussian noise channel, making use of the code's weight spectrum. In the present work, the bound is extended to memoryless symmetric channels with a discrete output alphabet. The derived bound is demonstrated on a hybrid BSC-BEC channel. Additionally, a reduced-complexity bound is introduced at the cost of some loss in tightness.

Extension of the Poltyrev Bound to Binary Memoryless Symmetric Channels

TL;DR

The paper extends the Poltyrev bound, a tight upper bound on decoding error probability, from its original BSC/AWGN settings to general binary memoryless symmetric channels with discrete outputs by employing the method of types. It derives a general, type-based bound that can be evaluated with polynomial-time complexity in the block length and demonstrates it on a hybrid BSC-BEC channel; a reduced-complexity version with linear-time cost is also presented. The approach yields explicit, channel-class-specific bounds for BSC, BEC, hybrid BSC-BEC, and a quinary channel, and shows significant improvements over existing bounds (e.g., Shulman-Feder, Miller-Burshtein) for moderate-length codes such as BCH codes. The results are particularly relevant for systems with small output alphabets and coarse quantization, such as certain flash-memory read channels, where accurate performance metrics are essential for code design and evaluation.

Abstract

The Poltyrev bound provides a very tight upper bound on the decoding error probability when using binary linear codes for transmission over the binary symmetric channel and the additive white Gaussian noise channel, making use of the code's weight spectrum. In the present work, the bound is extended to memoryless symmetric channels with a discrete output alphabet. The derived bound is demonstrated on a hybrid BSC-BEC channel. Additionally, a reduced-complexity bound is introduced at the cost of some loss in tightness.
Paper Structure (12 sections, 5 theorems, 28 equations, 8 figures)

This paper contains 12 sections, 5 theorems, 28 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Revisiting the Poltyrev Bound for the BSC
  4. Extension of the Poltyrev Bound to General Binary-Input Symmetric-Output Channels
  5. Channel Classes with Explicit Characterization of Error Event Regions
  6. Explicit Characterization for the Class of BSC
  7. Explicit Characterization for the Class of BEC
  8. Explicit Characterization for Hybrid BSC-BEC
  9. Explicit Characterization for a Class of Quinary Channels
  10. Computationally Simplified Bound
  11. Numerical Results
  12. Discussion

Key Result

Theorem 1

For a BMS channel eq:BMS, the error probability of a binary linear code with minimal Hamming distance $d_{\rm min}$ and weight distribution $S_w, w=d_{\rm min},\dots,n$, is bounded as follows

Figures (8)

  • Figure 1: Binary-input symmetric-output channel.
  • Figure 2: Binary symmetric channel - BSC($\varepsilon$).
  • Figure 3: Binary erasure channel - BEC($\delta$).
  • Figure 4: Hybrid BSC-BEC($\varepsilon,\delta$) channel.
  • Figure 5: Quinary($\varepsilon,\delta, \gamma$) channel.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Theorem 1: Binary Memoryless Symmetric Channel
  • proof
  • Remark 1
  • Lemma 1: BEC
  • Lemma 2: BSC-BEC
  • Lemma 3: Quinary channel
  • Corollary 1: Rectangle-type search region