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IEEE Transactions on Communications

On Girth Conditioning for Low-Density Parity-Check Codes

Samuele Bandi, Velio Tralli, Andrea Conti, Maddalena Nonato

Published Dec 22, 2010• 7 citations
Publisher

Abstract

Low-density parity-check (LDPC) codes are gaining interest for high data rate applications in both terrestrial and spatial communications. They can be designed and studied through a bipartite graph whose characteristics affect the performance. This paper proposes a low-complexity method to improve the performance of LDPC codes by selectively removing some cycles from the associated bipartite graph. The method is based on a modified version of the breadth first search (BFS) algorithm that we call modified BFS (MBFS), which is applied to find cycles, and a greedy procedure to eliminate them. Throughout the paper we will give a detailed description of the algorithm proposed and analytically study its complexity. Simulation results show that this girth conditioning method applied to some classes of codes, whose structure allows further optimization, can lead to a significative complexity reduction and a performance improvements with respect to other methods.

DOI
10.1109/tcomm.2011.121410.090017

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Low-density parity-check (LDPC) codes are gaining interest for high data rate applications in both terrestrial and spatial communications. They can be designed and studied through a bipartite graph whose characteristics affect the performance. This paper proposes a low-complexity method to improve the performance of LDPC codes by selectively removing some cycles from the associated bipartite graph. The method is based on a modified version of the breadth first search (BFS) algorithm that we call modified BFS (MBFS), which is applied to find cycles, and a greedy procedure to eliminate them. Throughout the paper we will give a detailed description of the algorithm proposed and analytically study its complexity. Simulation results show that this girth conditioning method applied to some classes of codes, whose structure allows further optimization, can lead to a significative complexity reduction and a performance improvements with respect to other methods.
This paper proposes a method to improve the performance of a low-density parity-check code, by selectively removing some cycles from the associated bipartite graph. The method is based on a modified version of breadth-first search (BFS) algorithm, that we call modified BFS (MBFS). Throughout the paper we will give a detailed description of the algorithm and analytically study its complexity. Simulation results will show that applying MBFS to a given code leads to significant improvement of its performance.
Low-density Parity check codes (LDPC) are gaining interest for high data rate application in both terrestrial and spatial communications which requires low bit error rate. This paper proposes an algorithm through which the best parity check matrix is obtained for irregular low density parity check codes, which reduces the Bit Error Rate ,Frame Error Rate and computational complexity. Tanner graph also called a bipartite graph format in an Low Density Parity Check (LDPC) code design contains many short cycles; it will produce a computational complexity in a code design process and it degrades the code performance. To overcome this effect, we proposed an new algorithm which obtains a best final parity check matrix. Throughout the paper we will give the detailed description of an efficient algorithm proposed. The complexity is studied graphically and analytically. Using proposed algorithm, cycles of length four observed in corresponding matrix are removed. Each matrix is evaluated over a noisy Additive White Gaussian Noise (AWGN) channel. Bit Error rate and Frame error rate is calculated. The results are compared with randomly generated best parity check matrix. Simulation results shows that proposed method successfully overcomes the computational complexity and meets the near Shannon limit.
It is well known that extremely long low-density parity-check (LDPC) codes perform exceptionally well for error correction applications, short-length codes are preferable in practical applications. However, short-length LDPC codes suffer from performance degradation owing to graph-based impairments such as short cycles, trapping sets and stopping sets and so on in the bipartite graph of the LDPC matrix. In particular, performance degradation at moderate to high Eb/N0 is caused by the oscillations in bit node a posteriori probabilities induced by short cycles and trapping sets in bipartite graphs. In this study, a computationally efficient algorithm is proposed to improve the performance of short-length LDPC codes at moderate to high Eb/N0. This algorithm makes use of the information generated by the belief propagation (BP) algorithm in previous iterations before a decoding failure occurs. Using this information, a reliability-based estimation is performed on each bit node to supplement the BP algorithm. The proposed algorithm gives an appreciable coding gain as compared with BP decoding for LDPC codes of a code rate equal to or less than 1/2 rate coding. The coding gains are modest to significant in the case of optimised (for bipartite graph conditioning) regular LDPC codes, whereas the coding gains are huge in the case of unoptimised codes. Hence, this algorithm is useful for relaxing some stringent constraints on the graphical structure of the LDPC code and for developing hardware-friendly designs.
This paper,based on algebra and graph,proposes a method for constructing structured low-density parity-check(LDPC) codes.In this method,the authors design a connected graph with three kinds of special path to ensure that the Tanner graph of the parity check matrix mapped from the connected graph is without short cycles.The construction method results in two classes of(3,k)-regular quasi-cyclic LDPC codes with girth 8 and 12 respectively.Furthermore,by extending this construction method,two classes of LDPC codes with column weight 2,whose girths are respectively 16 and 24,can be obtained.The simulation studies show that these codes can achieve better performance than randomly constructed regular LDPC codes over AWGN channels with iterative decoding.At the BER of 10-5,the new proposed(3,6)-regular(4356,2205) QC-LDPC code performs 1.48dB from the Shannon limit.Moreover,it outperforms the random regular LDPC code of comparable parameters by 0.1dB.For the new proposed structured LDPC codes with column weight 2,they can achieve about 2dB coding gains with respect to the random regular LDPC codes with similar block lengths and code rates.
This article describes the different methods to design regular low density parity-check (LDPC) codes with large girth. In graph terms, this corresponds to designing bipartite undirected regular graphs with large girth. Large girth speeds the convergence of iterative decoding and improves the performance at least in the high SNR range, by slowing down the onsetting of the error floor. We reviewed several existing constructions from exhaustive search to highly structured designs based on Euclidean and projective finite geometries and combinatorial designs. We describe GB and TS LDPC codes and compared the BER performance with large girth to the BER performance of random codes. These studies confirm that in the high SNR regime these codes with high girth exhibit better BER performance. The regularity of the codes provides additional advantages that we did not explore in this article like the simplicity of their hardware implementation and fast encoding.
Performance of medium length LDPC codes is directly proportional to error floor .In these codes, error floor can be mainly improved through elimination of short cycles.Hence, for effective elimination of short cycles, it is customary to use a girth conditioning in random construction process.Various accurate techniques, such as, density evolution and EXIT chart are developed for performance prediction of large block length, iteratively decoded LDPC codes.However, for medium block length, this method evolved with erroneous results.Even though, the medium block length LDPC codes have many application, short cycles degrades the performance and adds computational complexity in decoding process.