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IEEE Communications Letters

A Measure of the Girth Histogram for Improved Girth-Conditioning Construction of LDPC Codes

Hua Chen, Zhigang Cao

Published Apr 1, 2007• 1 citation
Publisher

Abstract

In this letter, we present a general quantitative measure of the girth histogram of low-density parity-check (LDPC) codes with joint consideration of the degrees and local girths of symbol nodes. We then propose a further girth-conditioning algorithm to optimize such a measure during the construction of LDPC codes. We show by simulation results that the algorithm improves not only the performance of irregular codes at high signal-to-noise ratios (SNR) considerably, but also the rates of regular codes

DOI
10.1109/lcom.2007.348294

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In this letter, we present a general quantitative measure of the girth histogram of low-density parity-check (LDPC) codes with joint consideration of the degrees and local girths of symbol nodes. We then propose a further girth-conditioning algorithm to optimize such a measure during the construction of LDPC codes. We show by simulation results that the algorithm improves not only the performance of irregular codes at high signal-to-noise ratios (SNR) considerably, but also the rates of regular codes
A low-density parity-check (LDPC) code can be described by a Tanner graph composed of symbol nodes and check nodes. Short cycles in the Tanner graph should be avoided because such cycles will degrade the performance of the decoder and hence increase the error rate. In most of the circumstances, cycles of length 4 can be totally eliminated in the Tanner graph when the LDPC code has been designed properly. Locally, for each variable node in the Tanner graph, the girth may take the value of 6 or 8, or even higher. In this study, the authors attempt to increase the local girths of a Tanner graph with an aim to improving the error performance of the code. The authors propose and analyse the ‘degree spectrum’ of the symbol nodes connecting to each of the check nodes. Based on the analysis, the authors can determine if the local girths can be further increased. The authors also propose simple ways to construct codes with higher local girths. Finally, the authors show the characteristics and the error performance of the codes constructed with the proposed methods.
Low Density Parity Check (LDPC) codes have capacity-approaching performance over several channels of interest. In practice, for good block-error rate performance, the girth of the Tanner graph of an LDPC code needs to be as high as possible. In theory, to show that block-error rate approaches zero for increasing block-lengths, the girth of the Tanner graph sequence needs to tend to infinity with block-length. To meet these requirements, we construct sequences of large-girth irregular LDPC codes for a given degree-distribution pair (DDP) by applying a general node splitting algorithm on large girth graphs. The obtained Tanner graph meets the required DDP up to a suitable approximation. By optimizing the node-splitting method and using suitable large-girth graphs, we show code constructions with smaller block length for the same girth, when compared to previous constructions. Similar gains in block length are observed in the construction of sequences of large-girth protograph LDPC codes. Simulations, over a binary erasure channel, confirm the gains in block-error rate obtained by the large girth construction.
The progressive-edge-growth (PEG) algorithm for constructing Tanner graphs with large girth by progressively establishing edges/connections between symbol nodes and check nodes in an edge-by-edge way is highly known to construct low-density parity-check (LDPC) codes at short block lengths for achieving good performance. This paper proposed an improved PEG construction for irregular codes, which heuristically selects good codes from random graphs for short block lengths. The simulation results are given to demonstrate that the proposed algorithm gians better BER performance for high signal-to-noise (SNR) ratios without performance loss for low-SNR region, i.e. low error floor.
In this paper, a method for constructing Low-Density Parity-Check (LDPC) codes of good performance in terms of Bit Error Rate (BER) using a computer search algorithm is presented. First, LDPC codes and encoding along with its Tanner graph representation is described. Then, a detailed description of the proposed construction algorithm is given. A Modified Shortest-Path (MSP) Algorithm of lower complexity is applied to obtain a code graph of the desired structure. The proposed algorithm works directly on the parity-check matrix of the LDPC code. The parity-check matrix of the code is constructed column by column subject to the bit-degree and girth constraints. Using the proposed algorithm, both regular and irregular LDPC codes have been constructed. Furthermore, not only have the girth been improved but also the rate of LDPC codes. Finally, by simulation results the constructed codes were shown to perform well over an additive white Gaussian noise (AWGN) channel.