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Decoding Error Probability of the Random Matrix Ensemble over the Erasure Channel

Chin Hei Chan, Fang-Wei Fu, Maosheng Xiong

TL;DR

This work analyzes the average decoding error probability of the random parity-check matrix ensemble $\mathcal{R}_{m,n}$ over a $q$-ary erasure channel under unambiguous, list, and maximum-likelihood decoding. It derives explicit formulas and error exponents for all three decoding principles, and proves a strong concentration result for unambiguous decoding showing ensemble performance closely tracks the average as the code length grows. The results are aligned with the behavior of the random $[n,n-m]_q$ code ensemble, with the ensemble structure enabling explicit counting via Gaussian $q$-binomial coefficients. The paper also provides a detailed variance analysis and demonstrates WHP convergence under certain rate conditions, highlighting practical implications for code design in erasure environments.

Abstract

Using tools developed in a recent work by Shen and the second author, in this paper we carry out an in-depth study on the average decoding error probability of the random matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas for the average decoding error probabilities of the random matrix ensemble under these three decoding principles and compute the error exponents. Moreover, for unambiguous decoding, we compute the variance of the decoding error probability of the random matrix ensemble and the error exponent of the variance, which imply a strong concentration result, that is, roughly speaking, the ratio of the decoding error probability of a random code in the ensemble and the average decoding error probability of the ensemble converges to 1 with high probability when the code length goes to infinity.

Decoding Error Probability of the Random Matrix Ensemble over the Erasure Channel

TL;DR

This work analyzes the average decoding error probability of the random parity-check matrix ensemble over a -ary erasure channel under unambiguous, list, and maximum-likelihood decoding. It derives explicit formulas and error exponents for all three decoding principles, and proves a strong concentration result for unambiguous decoding showing ensemble performance closely tracks the average as the code length grows. The results are aligned with the behavior of the random code ensemble, with the ensemble structure enabling explicit counting via Gaussian -binomial coefficients. The paper also provides a detailed variance analysis and demonstrates WHP convergence under certain rate conditions, highlighting practical implications for code design in erasure environments.

Abstract

Using tools developed in a recent work by Shen and the second author, in this paper we carry out an in-depth study on the average decoding error probability of the random matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas for the average decoding error probabilities of the random matrix ensemble under these three decoding principles and compute the error exponents. Moreover, for unambiguous decoding, we compute the variance of the decoding error probability of the random matrix ensemble and the error exponent of the variance, which imply a strong concentration result, that is, roughly speaking, the ratio of the decoding error probability of a random code in the ensemble and the average decoding error probability of the ensemble converges to 1 with high probability when the code length goes to infinity.

Paper Structure

This paper contains 14 sections, 12 theorems, 121 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Statement of the main results
  4. Preliminaries
  5. Three decoding principles, the average decoding error probability, and the error exponent
  6. Gaussian binomial coefficients
  7. Three counting results for the ensemble $\mathcal{R}_{m,n}$
  8. Proof of Theorem \ref{['Exp']}
  9. Proof of Theorem \ref{['Errexp']}
  10. Proof of Theorem \ref{['Concen']}
  11. The variance of unsuccessful decoding probability and its error exponent
  12. Proof of Theorem \ref{['Concen']}
  13. Appendix: Proof of Theorem \ref{['Errexpvar']}
  14. Conclusion

Key Result

theorem 1

Let $\mathcal{R}_{m,n}$ be the random matrix ensemble described above. Denote by $\genfrac{[}{]}{0pt}{}{i}{j}_q$ the Gaussian $q$-binomial coefficient and denote

Figures (2)

  • Figure 1: The error exponent $T_{\mathrm{ld}}(\ell,\varepsilon)$ for $0<R<1$ where $\ell=0,1,2$ and $q=2, \varepsilon=0.25$.
  • Figure 2: The error exponent $S_{\mathrm{ud}}(\varepsilon)$ for $0<R<1$, where $q=2, \varepsilon=0.25$.

Theorems & Definitions (19)

  • theorem 1
  • remark thmcounterremark
  • theorem 2
  • theorem 3
  • lemma thmcounterlemma: MacWilliams
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • ...and 9 more