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International Symposium on Turbo Codes and Iterative Information Processing

A Low Complexity PEG-like Algorithm to Construct Quasi-Cyclic LDPC Codes

Anthony Gómez-Fonseca, Roxana Smarandache, David G. M. Mitchell

Published Sep 4, 2023• 4 citations
Publisher

Abstract

In this paper, we introduce a modified version of the Progressive Edge-Growth (PEG) algorithm that is used to construct the Tanner graphs of quasi-cyclic (QC) low-density parity-check (LDPC) codes having large girth. In our algorithms, we replace certain requirements in the original PEG algorithm, namely the expansion of subgraphs from symbol nodes and the identification of a most distant check node before placing a new edge, by the construction of certain minimal sets, called forbidden sets, that contain the problematic quasi-cyclic constraints that do not let us exceed a given girth. We propose some exponent-selection strategies that are shown to increase the chance of obtaining large girth, and reduce the number of short cycles in the Tanner graph.

DOI
10.1109/istc57237.2023.10273481

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In this paper, we introduce a modified version of the Progressive Edge-Growth (PEG) algorithm that is used to construct the Tanner graphs of quasi-cyclic (QC) low-density parity-check (LDPC) codes having large girth. In our algorithms, we replace certain requirements in the original PEG algorithm, namely the expansion of subgraphs from symbol nodes and the identification of a most distant check node before placing a new edge, by the construction of certain minimal sets, called forbidden sets, that contain the problematic quasi-cyclic constraints that do not let us exceed a given girth. We propose some exponent-selection strategies that are shown to increase the chance of obtaining large girth, and reduce the number of short cycles in the Tanner graph.
For a given base graph, the protograph can be obtained by a copy-and-permute procedure. If the permutation is cyclic, the protograph corresponds to a quasi-cyclic (QC) protograph LDPC code. The girth of the QC protograph LDPC code is determined by the girth of the base graph and the permutation shifts. Progressive edge-growth (PEG) construction builds up a Tanner graph, or equivalently a parity-check matrix, for an LDPC code by maximizing the local girth at symbol nodes in a greedy algorithm. In this paper, we introduce a modified version of the PEG algorithm which can construct large girth base graph and determine the optimal permutation shifts, simultaneously, for QC protograph LDPC codes. Simulation results show that the QC protograph LDPC codes constructed by the proposed modified PEG algorithm have good frame error rate (FER) performance over the AWGN channel.
Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.
The Progressive edge-growth (PEG) algorithm was proven to be a simple and effective approach in designing Low-density parity-check (LDPC) codes. However, the Tanner graph constructed with the PEG algorithm is non-structured, which leads to random distributions of the positions of 1's in the corresponding parity-check matrix. In this paper, an improved algorithm based on the PEG algorithm for constructing quasi-cyclic (QC) LDPC codes is proposed. The proposed algorithm can eliminate short cycles efficiently and gain low error-floors. What is more important, the proposed QC-LDPC codes construction method is hardware-friendly because of its quasi-cyclic structure. Our simulation results demonstrate that the codes constructed with the proposed algorithm has better performance than the PEG algorithm over the additive white Gaussian noise (AWGN) channel and Rayleigh flat fading channel.
In this paper, we propose an optimized belief propagation (OBP) based progressive edge-growth (PEG) method for constructing quasi-cyclic low density parity check (QC-LDPC) codes. In this proposed method, Tanner graphs are built by progressively appending the check nodes rather than the variable nodes. Moreover, this OBP-PEG method considers three new constraint conditions to select the QC-LDPC code sets constructed by the PEG method. Compared with the PEG-based QC-LDPC decoders, the proposed OBP-PEG decoders decrease the number of the input ports of check-node processer by up to 25%, accelerates the convergence speed by up to 11.7% in layered decoding process, and improves the success probability by up to ten times in constructing fast-convergence QC-LDPC codes.