1. Improved Progressive-Edge-Growth (PEG) Construction of Irregular LDPC CodesBy
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 k=0 to dsj -1 do { if k=0 {
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 \ Nℓ+1sj = 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 \ Nℓ+1sj = Ø but C \ Nℓsj ≠ Ø; Set of candidate check nodes (with lowest degree) : Ωksj

6. Modified PEG Algorithm
The addition of Eksj to the graph creates new cycles all with length 2(ℓ+2)
We select a check node from Ωksj 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:
undetected errors
dmin
Trapping sets (w,u)
PEG
36% 12
(6,1) and many with u=1, w ≤13
M-PEG
10% 15
(15,1)

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