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

Outline Introduction and Motivation Improved PEG Algorithm Simulation Results Concluding Remarks

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

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

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

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

Simulation Results

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)

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