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Rate-Adaptive Protograph-Based MacKay-Neal Codes

Ayman Zahr, Emna Ben Yacoub, Balázs Matuz, Gianluigi Liva

TL;DR

This work develops rate-adaptive Protograph MacKay-Neal (MN) codes by concatenating an outer distribution matcher (DM) with an inner protograph LDPC code, achieving variable rates at fixed blocklengths. An equivalent parallel-channel (EPC) model permits tractable analysis of the nonlinear MN construction, enabling density evolution (via PEXIT and quantized DE) and distance-spectrum techniques to predict both waterfall and error-floor behavior. A distance-spectrum-based criterion is introduced to filter out MN ensembles prone to high error floors, and a design methodology using the WCL measure demonstrates that near-capacity performance (within ~1 dB of the Shannon limit) is achievable across a broad range of rates with a single inner LDPC code. The results offer a practical pathway to high-throughput, rate-flexible transmission in binary-input systems, leveraging a fixed inner code and a tunable outer DM.

Abstract

Rate-adaptive MacKay-Neal (MN) codes based on protographs are analyzed. The code construction employs an outer distribution matcher (DM) to adapt the rate of the scheme. The DM is coupled with an inner protograph-based low-density parity-check (LDPC) code. The performance achievable by the resulting code structure, that is nonlinear, is studied by means of an equivalent communication model that reduces the problem to the analysis of the inner (linear) LDPC code with transmission that takes place in parallel over the communication channel, and over a suitably defined binary symmetric channel. A density evolution analysis of protograph MN code ensembles is outlined, and it is complemented by an error floor analysis that relies on the derivation of the average input-output weight distribution of the inner LDPC code ensemble. Conditions on the shape of the normalized logarithmic asymptotic input-output weight distribution are defined, which allow discarding code ensembles with bad error floor properties during the code design phase. Examples of code designs are provided, showing how the use of a single LDPC code ensemble allows operating within 1 dB from the Shannon limit over a wide range of code rates, where the code rate is selected by tuning the DM parameters. By enabling rate flexibility with a constant blocklength, and with a fixed LDPC code as inner code, the construction provides an appealing solution for very high-throughput wireless (optical) links that employ binary-input modulations.

Rate-Adaptive Protograph-Based MacKay-Neal Codes

TL;DR

This work develops rate-adaptive Protograph MacKay-Neal (MN) codes by concatenating an outer distribution matcher (DM) with an inner protograph LDPC code, achieving variable rates at fixed blocklengths. An equivalent parallel-channel (EPC) model permits tractable analysis of the nonlinear MN construction, enabling density evolution (via PEXIT and quantized DE) and distance-spectrum techniques to predict both waterfall and error-floor behavior. A distance-spectrum-based criterion is introduced to filter out MN ensembles prone to high error floors, and a design methodology using the WCL measure demonstrates that near-capacity performance (within ~1 dB of the Shannon limit) is achievable across a broad range of rates with a single inner LDPC code. The results offer a practical pathway to high-throughput, rate-flexible transmission in binary-input systems, leveraging a fixed inner code and a tunable outer DM.

Abstract

Rate-adaptive MacKay-Neal (MN) codes based on protographs are analyzed. The code construction employs an outer distribution matcher (DM) to adapt the rate of the scheme. The DM is coupled with an inner protograph-based low-density parity-check (LDPC) code. The performance achievable by the resulting code structure, that is nonlinear, is studied by means of an equivalent communication model that reduces the problem to the analysis of the inner (linear) LDPC code with transmission that takes place in parallel over the communication channel, and over a suitably defined binary symmetric channel. A density evolution analysis of protograph MN code ensembles is outlined, and it is complemented by an error floor analysis that relies on the derivation of the average input-output weight distribution of the inner LDPC code ensemble. Conditions on the shape of the normalized logarithmic asymptotic input-output weight distribution are defined, which allow discarding code ensembles with bad error floor properties during the code design phase. Examples of code designs are provided, showing how the use of a single LDPC code ensemble allows operating within 1 dB from the Shannon limit over a wide range of code rates, where the code rate is selected by tuning the DM parameters. By enabling rate flexibility with a constant blocklength, and with a fixed LDPC code as inner code, the construction provides an appealing solution for very high-throughput wireless (optical) links that employ binary-input modulations.
Paper Structure (21 sections, 4 theorems, 55 equations, 12 figures, 2 tables)

This paper contains 21 sections, 4 theorems, 55 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Channel Model
  4. Protograph LDPC Codes
  5. Asymptotic Enumeration Results
  6. Protograph MacKay-Neal Codes
  7. Protograph-based Construction
  8. Decoding of MN Codes
  9. Belief Propagation Decoding
  10. Maximum-likelihood and Mismatched Decoding
  11. Equivalent Parallel Channel Model
  12. Density Evolution Analysis
  13. Accuracy of the PEXIT Analysis
  14. Distance Spectrum Analysis
  15. Union Bound under Mismatched Decoding
  16. ...and 6 more sections

Key Result

Lemma 1

[Hayman Formula for Multivariate Polynomials DUR06 Let $\bm{z} = ( z_{1}, z_{2},\ldots, z_{d})$ and let $p (\bm{z})$ be a multivariate polynomial with $p(\bm{0}) \neq 0$. Let $\bm{\beta} = ( \beta_{1}, \beta_{2}, \ldots, \beta_{d})$ where $0 \leq \beta_{t} \leq 1$ and $\beta_{t}n$ is an integer for where $\mathop{\mathrm{coeff}}\nolimits\left( p(\bm{z}) ^{n} , \bm{z} ^{n \bm{\beta}} \right)$ rep

Figures (12)

  • Figure 1: Protograph of Example \ref{['ex:proto']}.
  • Figure 2: Tanner graph obtained by lifting the protograph of Example \ref{['ex:proto']}.
  • Figure 3: System model, where a MN code is used to communicate over the biAWGN channel (communication channel).
  • Figure 4: Belief propagation decoding over the Tanner graph of the mother LDPC code.
  • Figure 5: Modification of the system model of Figure \ref{['fig:model']}, where an i.i.d. scrambler is introduced.
  • ...and 7 more figures

Theorems & Definitions (18)

  • Example 1
  • Lemma 1
  • Remark 1
  • Example 2
  • Example 3
  • Remark 2
  • Lemma 2
  • Theorem 1
  • Example 4
  • Lemma 3
  • ...and 8 more