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On the Error Rate of Binary BCH Codes under Error-and-erasure Decoding

Sisi Miao, Jonathan Mandelbaum, Holger Jäkel, Laurent Schmalen

TL;DR

The paper tackles the challenge of estimating decoding error probabilities for binary BCH codes under error-and-erasure decoding by deriving a closed-form decoding transition probability for EaED, $P(R=r|U=u,E=e)$, and validating it via simulations. It introduces anchor-bit extensions to mitigate miscorrections, resulting in EaED$^{\mathsf{a}}$ with significantly improved one-shot decoding performance. The work further demonstrates substantial post-FEC FER gains in concatenated BCH-RS schemes when replacing BDD with EaED$^{\mathsf{a}}$, while keeping complexity modest. These contributions enable accurate performance prediction for soft-information decoding of BCH codes and offer practical guidance for high-throughput communication systems.

Abstract

Determining the exact decoding error probability of linear block codes is an interesting problem. For binary BCH codes, McEliece derived methods to estimate the error probability of a simple bounded distance decoding (BDD) for BCH codes. However, BDD falls short in many applications. In this work, we consider error-and-erasure decoding and its improved variants. We derive closed-form expressions for their error probabilities and validate them through simulations. Then, we illustrate their use in assessing concatenated coding schemes.

On the Error Rate of Binary BCH Codes under Error-and-erasure Decoding

TL;DR

The paper tackles the challenge of estimating decoding error probabilities for binary BCH codes under error-and-erasure decoding by deriving a closed-form decoding transition probability for EaED, , and validating it via simulations. It introduces anchor-bit extensions to mitigate miscorrections, resulting in EaED with significantly improved one-shot decoding performance. The work further demonstrates substantial post-FEC FER gains in concatenated BCH-RS schemes when replacing BDD with EaED, while keeping complexity modest. These contributions enable accurate performance prediction for soft-information decoding of BCH codes and offer practical guidance for high-throughput communication systems.

Abstract

Determining the exact decoding error probability of linear block codes is an interesting problem. For binary BCH codes, McEliece derived methods to estimate the error probability of a simple bounded distance decoding (BDD) for BCH codes. However, BDD falls short in many applications. In this work, we consider error-and-erasure decoding and its improved variants. We derive closed-form expressions for their error probabilities and validate them through simulations. Then, we illustrate their use in assessing concatenated coding schemes.

Paper Structure

This paper contains 14 sections, 2 theorems, 23 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Analysis of Bounded Distance Decoding
  4. Error-and-erasure Decoding
  5. Error-and-erasure (EaE) Decoding Algorithm
  6. Decoding Transition Probability
  7. Miscorrection Reduction with Anchor Bits
  8. BDD with Anchor Bits
  9. EaED with Anchor Bits
  10. Improving one-shot decoding of BCH code
  11. Application in Concatenated BCH-RS Schemes
  12. Conclusion
  13. EaED Calculation
  14. EaED with Anchor Bits

Key Result

Theorem 1

The DTP of BDD is given as follows. For $u \le t$, $P_{\textnormal{BDD}}(0|u) = 1$, $P_{\textnormal{BDD}}(r|u) = 0$, $(d,r)\neq (\mathsf{succ},0)$. For $u > t$,

Figures (5)

  • Figure 1: Graphical illustration of the miscorrection scenario when using a BDD.
  • Figure 2: Simulated and computed BER results of the $[255,239,5]$ primitive BCH code. The $\mathsf{EaED}$ uses the parameter $\mathsf{T} = 0.16$ and the ${\textnormal{EaED}}^{\textsf{a}}$ uses the parameters $\mathsf{T} = 0.13$ and $\mathsf{T}_{\textnormal{a}}=0.75$. The OSD uses order 2, as higher orders do not yield additional performance gains.
  • Figure 3: FER versus crossover probability for eight RS$[544, 514, 15]$ outer code interleaved with $64$ BCH$[700, 680, 2]$ inner code. The EaED$^{\mathsf{a}}$ uses the parameters $\mathsf{T} = 0.05$ and $\mathsf{T}_{\textnormal{a}}=0.56$ which minimizes the FER at $7$ dB.
  • Figure 4: Sorting of the cases in Tab. \ref{['tab:BDDoutcomes']} according to the value of $e_1$. When $u>t$, sets $\mathcal{L}$ and $\mathcal{R}$ will not occur.
  • Figure 5: Graphical illustration of case ➁. Orange parts sum up to $e_1$ while green parts sum up to $e_2=e-e_1$. The gray parts sum up to $d_{\sim{\textnormal{E}(\bm{y})}}(\bm{c})$.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • proof