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Systematic Bernoulli Generator Matrix Codes

Yixin Wang, Fanhui Meng, Xiao Ma

TL;DR

The paper proves that systematic BGM codes achieve capacity on BIOS channels in terms of FER, using a novel framework that handles systematic encoders without requiring pairwise independence. It also shows Bernoulli parity-check (BPC) codes are capacity-achieving within the same framework, by relating inner parity-check transmission to an effective systematic code. The authors derive finite-length BER/FER lower bounds via IOWEF and a list-coset decoding analysis, and validate with simulations that match error floors; they further improve water-fall performance through a statistical-physics-inspired graph design and enhance the error floor by outer-code concatenation. The work offers practical, sparse, systematic codes with provable capacity properties and practical tuning via graph assortativity and concatenation.

Abstract

This paper is concerned with the systematic Bernoulli generator matrix~(BGM) codes, which have been proved to be capacity-achieving over binary-input output-symmetric~(BIOS) channels in terms of bit-error rate~(BER). We prove that the systematic BGM codes are also capacity-achieving over BIOS channels in terms of frame-error rate (FER). To this end, we present a new framework to prove the coding theorems for binary linear codes. Different from the widely-accepted approach via ensemble enlargement, the proof directly applies to the systematic binary linear codes. The new proof indicates that the pair-wise independence condition is not necessary for proving the binary linear code ensemble to achieve the capacity of the BIOS channel. The Bernoulli parity-check~(BPC) codes, which fall within the framework of the systematic BGM codes with parity-check bits known at the decoder can also be proved to achieve the capacity. The presented framework also reveals a new mechanism pertained to the systematic linear codes that the systematic bits and the corresponding parity-check bits play different roles. Precisely, the noisy systematic bits are used to limit the list size of candidate codewords, while the noisy parity-check bits are used to select from the list the maximum likelihood codeword. For systematic BGM codes with finite length, we derive the lower bounds on the BER and FER, which can be used to predict the error floors. Numerical results show that the systematic BGM codes match well with the derived error floors. The performance in water-fall region can be improved with approaches in statistical physics and the error floors can be significantly improved by implementing the concatenated codes with the systematic BGM codes as the inner codes.

Systematic Bernoulli Generator Matrix Codes

TL;DR

The paper proves that systematic BGM codes achieve capacity on BIOS channels in terms of FER, using a novel framework that handles systematic encoders without requiring pairwise independence. It also shows Bernoulli parity-check (BPC) codes are capacity-achieving within the same framework, by relating inner parity-check transmission to an effective systematic code. The authors derive finite-length BER/FER lower bounds via IOWEF and a list-coset decoding analysis, and validate with simulations that match error floors; they further improve water-fall performance through a statistical-physics-inspired graph design and enhance the error floor by outer-code concatenation. The work offers practical, sparse, systematic codes with provable capacity properties and practical tuning via graph assortativity and concatenation.

Abstract

This paper is concerned with the systematic Bernoulli generator matrix~(BGM) codes, which have been proved to be capacity-achieving over binary-input output-symmetric~(BIOS) channels in terms of bit-error rate~(BER). We prove that the systematic BGM codes are also capacity-achieving over BIOS channels in terms of frame-error rate (FER). To this end, we present a new framework to prove the coding theorems for binary linear codes. Different from the widely-accepted approach via ensemble enlargement, the proof directly applies to the systematic binary linear codes. The new proof indicates that the pair-wise independence condition is not necessary for proving the binary linear code ensemble to achieve the capacity of the BIOS channel. The Bernoulli parity-check~(BPC) codes, which fall within the framework of the systematic BGM codes with parity-check bits known at the decoder can also be proved to achieve the capacity. The presented framework also reveals a new mechanism pertained to the systematic linear codes that the systematic bits and the corresponding parity-check bits play different roles. Precisely, the noisy systematic bits are used to limit the list size of candidate codewords, while the noisy parity-check bits are used to select from the list the maximum likelihood codeword. For systematic BGM codes with finite length, we derive the lower bounds on the BER and FER, which can be used to predict the error floors. Numerical results show that the systematic BGM codes match well with the derived error floors. The performance in water-fall region can be improved with approaches in statistical physics and the error floors can be significantly improved by implementing the concatenated codes with the systematic BGM codes as the inner codes.

Paper Structure

This paper contains 25 sections, 10 theorems, 57 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. A New Framework
  3. Problem Statement
  4. Coding Theorem for Systematic Linear Codes
  5. Main Results
  6. Coding Theorem for Systematic BGM Codes
  7. Coding Theorem for Bernoulli Parity-check Codes
  8. Performance of Finite Length BGM Codes
  9. Proof of the Coding Theorems
  10. Partial Mutual Information
  11. Partial Error Exponent
  12. List Coset Decoding
  13. Proof of Theorem \ref{['main_theorem']}
  14. Proof of Theorem \ref{['LDPC_theorem']}
  15. Finite Length Performance of Systematic BGM Codes
  16. ...and 10 more sections

Key Result

Theorem 1

Consider systematic binary linear block codes defined by the generator matrices of the form $[\mathbf{I}~\mathbf{G}]$, where $\mathbf{I}$ is the identity matrix of order $k$ and $\mathbf{G}$ is a binary matrix of size $k\times m$. For two arbitrarily small positive numbers $\epsilon$ and $\delta$, o

Figures (8)

  • Figure 1: A system model with systematic linear coding.
  • Figure 2: The BER and FER performance of systematic BGM codes with $k=1024$ and $m=1024$. The parameter $\rho$ is the density of the Bernoulli random matrix $\mathbf{G}$ and "$w = 8$" is for the fixed row weight of a random matrix $\mathbf{G}$ constructed by Gallager's approach.
  • Figure 3: An illustration of different node interaction patterns. Here, we take the BGM code with $k=1024, m=1024, \rho=0.01$ as an example. Top: the disassortative (a), random (b), and assortative (c) node interaction patterns. Bottom: the corresponding joint degree distributions.
  • Figure 4: The BER performance of systematic BGM codes of $k=1024$, $m=1024$ and $\rho=0.01$ with different assortativity coefficient $r$. The black dashed line represents the lower bound.
  • Figure 5: Average number of iterations of systematic BGM codes of $k=1024$, $m=1024$ and $\rho=0.01$ with different assortativity coefficient $r$.
  • ...and 3 more figures

Theorems & Definitions (15)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • Lemma 2
  • Theorem 6
  • Theorem 7
  • ...and 5 more