Improved ProgressiveEdgeGrowth PEG Construction of Irregular LDPC Codes

1
Improved Progressive-Edge-Growth (PEG)
Construction of Irregular LDPC Codes

• By
• Hua Xiao and Amir H. Banihashemi
• Department of Systems and Computer Engineering
• Broadband Communications and Wireless Systems
  (BCWS) Centre
• Carleton University
• Ottawa, Ontario, Canada

2
Outline

• Introduction and Motivation
• Improved PEG Algorithm
• Simulation Results
• Concluding Remarks

3
Introduction and Motivation

• Construction of good LDPC codes at short and
  intermediate block lengths is of great practical
  importance
• Progressive-Edge-Growth (PEG), proposed by Hu,
  Eleftheriou and Arnold in 2001, is among the
  best
• - Constructs the Tanner graph edge-by-edge
  by maximizing the local girth at variable nodes
  in a greedy fashion
• - Simple and flexible
• - Linear-time encodable codes
• - Both regular and irregular
• For irregular codes, PEG with optimized variable
  node degree distributions result in very good
  performance, especially in the waterfall region
• The good performance in the waterfall region is
  usually counter-balanced by a relatively poor
  performance in the error-floor region

4

• Goal Improve the performance of irregular PEG at
  high SNR region without any performance
  degradation in low SNR region
• Main idea
• - Problem Short cycles are not good for
  iterative decoding
• - Solution When there are more than one
  candidate check nodes to be connected to a
  variable node, choose the one that provides the
  highest degree of connectivity for the newly
  created cycles to the rest of the graph

5
PEG Algorithm

• for j 0 to n -1 do
• for k0 to dsj -1 do
• if k0
• connect sj to a check node that has the
  lowest degree under the current graph setting
• else
• expand a subgraph from sj up to depth l under
  the current graph setting such that Nlsj stops
  increasing but is less than m, or the C \ Nl1sj
  Ø but C \ Nlsj ? Ø, then connect the k-th edge
  of sj (Eksj) to a check node picked from the set
  C \ Nlsj which has the lowest degree.
• Our focus k 1, C \ Nl1sj Ø but C \ Nlsj ?
  Ø Set of candidate check nodes (with lowest
  degree) Oksj

6
Modified PEG Algorithm

• The addition of Eksj to the graph creates new
  cycles all with length 2(l2)
• We select a check node from Oksj whose associated
  cycles have the highest degree of connectivity to
  the rest of the graph
• Measure of connectivity Approximate Cycle
  Extrinsic message degree (ACE) ?i(di-2) Tian
  et al., 2003
• We maximize the minimum ACE for the new cycles

7
Simulation Results
8
Simulation Results

• At high-SNR, errors are due to low-weight
  codewords (undetected errors) and low-weight
  trapping sets Richardson, 2003 or
  near-codewords Mackay and Postol, 2003 with
  small number of unsatisfied checks (detected
  errors)
• For (1008,504) at 2.8 dB

9
Concluding Remarks

• Irregular LDPC codes constructed by PEG based on
  optimal variable-node degree sequences perform
  very well in the waterfall region
• The performance at higher SNR values however is
  usually not as good and can be impaired by an
  early error floor
• we propose a very simple modification to PEG
  algorithm which considerably enhances the
  performance at high SNR region without any
  degradation in low-SNR performance
• The modification is based on creating a higher
  degree of connectivity in the Tanner graph of the
  code without sacrificing the girth distribution
• This appears to improve both the minimum distance
  and the trapping sets (near-codewords) of the
  code