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LDPC Codes Achieve List Decoding Capacity

Jonathan Mosheiff, Nicolas Resch, Noga Ron-Zewi, Shashwat Silas, Mary Wootters

TL;DR

This work proves that Gallager’s LDPC code ensembles achieve list-decoding capacity with high probability, the first graph-based family to do so. The authors introduce a general local-property framework and show a reduction: any local property that random linear codes satisfy with high probability is also satisfied by random $s$-LDPC codes at nearly the same rate, provided $s$ is sufficiently large. They establish sharp thresholds for local properties of random linear codes via an implied-distribution construction, and they show that LDPC codes inherit these properties by a Fourier-analytic argument that compares containment probabilities to the linear-code baseline. They also prove that random $s$-LDPC codes attain the Gilbert-Varshamov bound in relative distance, strengthening the link between LDPC structure and capacity-achieving capabilities. Overall, the results open the possibility of truly linear-time list-decoding for capacity-achieving codes by leveraging graph-based LDPC constructions.

Abstract

We show that Gallager's ensemble of Low-Density Parity Check (LDPC) codes achieves list-decoding capacity with high probability. These are the first graph-based codes shown to have this property. This result opens up a potential avenue towards truly linear-time list-decodable codes that achieve list-decoding capacity. Our result on list decoding follows from a much more general result: any $\textit{local}$ property satisfied with high probability by a random linear code is also satisfied with high probability by a random LDPC code from Gallager's distribution. Local properties are properties characterized by the exclusion of small sets of codewords, and include list-decodability, list-recoverability and average-radius list-decodability. In order to prove our results on LDPC codes, we establish sharp thresholds for when local properties are satisfied by a random linear code. More precisely, we show that for any local property $\mathcal{P}$, there is some $R^*$ so that random linear codes of rate slightly less than $R^*$ satisfy $\mathcal{P}$ with high probability, while random linear codes of rate slightly more than $R^*$, with high probability, do not. We also give a characterization of the threshold rate $R^*$.

LDPC Codes Achieve List Decoding Capacity

TL;DR

This work proves that Gallager’s LDPC code ensembles achieve list-decoding capacity with high probability, the first graph-based family to do so. The authors introduce a general local-property framework and show a reduction: any local property that random linear codes satisfy with high probability is also satisfied by random -LDPC codes at nearly the same rate, provided is sufficiently large. They establish sharp thresholds for local properties of random linear codes via an implied-distribution construction, and they show that LDPC codes inherit these properties by a Fourier-analytic argument that compares containment probabilities to the linear-code baseline. They also prove that random -LDPC codes attain the Gilbert-Varshamov bound in relative distance, strengthening the link between LDPC structure and capacity-achieving capabilities. Overall, the results open the possibility of truly linear-time list-decoding for capacity-achieving codes by leveraging graph-based LDPC constructions.

Abstract

We show that Gallager's ensemble of Low-Density Parity Check (LDPC) codes achieves list-decoding capacity with high probability. These are the first graph-based codes shown to have this property. This result opens up a potential avenue towards truly linear-time list-decodable codes that achieve list-decoding capacity. Our result on list decoding follows from a much more general result: any property satisfied with high probability by a random linear code is also satisfied with high probability by a random LDPC code from Gallager's distribution. Local properties are properties characterized by the exclusion of small sets of codewords, and include list-decodability, list-recoverability and average-radius list-decodability. In order to prove our results on LDPC codes, we establish sharp thresholds for when local properties are satisfied by a random linear code. More precisely, we show that for any local property , there is some so that random linear codes of rate slightly less than satisfy with high probability, while random linear codes of rate slightly more than , with high probability, do not. We also give a characterization of the threshold rate .

Paper Structure

This paper contains 31 sections, 15 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. List Decoding.
  3. LDPC Codes.
  4. Motivating question.
  5. Contributions
  6. (1) Random LDPC codes achieve list-decoding capacity.
  7. (2) Random LDPC codes achieve any local property that random linear codes achieve.
  8. (3) A characterization of local properties satisfied by random linear codes.
  9. Related Work
  10. List-decodability of random ensembles of codes.
  11. List-decodability of explicit codes.
  12. LDPC Codes Achieve Capacity on the Binary Symmetric Channel.
  13. Relationship to threshold results in combinatorics.
  14. Discussion and open questions
  15. Organization and main building blocks
  16. ...and 16 more sections

Key Result

Theorem 1.2

For any $R \in (0,1)$, $\varepsilon>0$, prime power $q$, $\alpha \in (0,1-1/q)$ and $L \geq1$ there exists $s_0 = s_0(\varepsilon,\alpha,q,L) \geq 1$ such that the following holds for any odd $s \geq s_0$. Suppose that a random linear code of rate $R$ over ${\mathbb F}_q$ is $(\alpha,L)$-list decoda

Figures (2)

  • Figure 1: A random $(t,s)$-regular bipartite graph that gives rise to a random $s$-LDPC code of rate $R$. Here, we set $t:=s(1-R)$.
  • Figure 2: The matrices $F$ and $H$. Each layer $H_i$ of $H$ is drawn independently according to the distribution $F \cdot \Pi \cdot D$, where $\Pi \in \{0,1\}^{n \times n}$ is a random permutation and $D \in {\mathbb F}_q^{n \times n}$ is a diagonal matrix with diagonal entries that are uniform in ${\mathbb F}_q^*$.

Theorems & Definitions (68)

  • Theorem 1.2
  • Remark 1.3: The parity of $s$
  • Remark 1.4: Dependence of $s_0$
  • Corollary 1.5
  • Remark 1.6: Other parameter regimes
  • Definition 1.7: Local property
  • Remark 1.8
  • Theorem 1.9: Main
  • Remark 1.10: The dependence on $\varepsilon, \bar{R}, q, b$
  • Remark 1.11: A converse to Theorem \ref{['thm:main']}?
  • ...and 58 more