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Alternative good LDPC codes for DVB-S2

Yang Xiao, Kiseon Kim

Published Oct 1, 2008• 13 citations
Publisher

Abstract

This paper reveals that the current LDPC codes in DVB-S2 are not good, since they have the codes of minimum weight of 5, which leads to their BER can not satisfy the requirements of DVB-S2. This paper develops an algorithm to estimate the minimum weight of LDPC codes based on generator matrices. Using it, we can find the minimum weight problem of the current LDPC codes in DVB-S2. To enhance the minimum weight problem of the LDPC codes, we propose a new approach to construct the encodable irregular QC codes, which can not produce new short girths and can have much larger minimum weight and minimum distance of the codes. Simulations verify the construction of alternative QC LDPC codes for DVB-S2 to be valid, since better BER performance than that of existing irregular LDPC codes of DVB-S2 is obtained.

DOI
10.1109/icosp.2008.4697527

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This paper reveals that the current LDPC codes in DVB-S2 are not good, since they have the codes of minimum weight of 5, which leads to their BER can not satisfy the requirements of DVB-S2. This paper develops an algorithm to estimate the minimum weight of LDPC codes based on generator matrices. Using it, we can find the minimum weight problem of the current LDPC codes in DVB-S2. To enhance the minimum weight problem of the LDPC codes, we propose a new approach to construct the encodable irregular QC codes, which can not produce new short girths and can have much larger minimum weight and minimum distance of the codes. Simulations verify the construction of alternative QC LDPC codes for DVB-S2 to be valid, since better BER performance than that of existing irregular LDPC codes of DVB-S2 is obtained.
The existing encodable irregular LDPC codes may have a degraded BER performance when it is derived from classical regular QC LDPC codes since both girth property and weight property are not carefully given attention to two or more thing, which lead to some codes with large girths have small minimum weights and poor BER performance. To solve the problem, we propose a new approach to construct the encodable irregular QC codes, which can not produce new short girths and can keep the large minimum weight and minimum distance of the codes. The theorems of girth 4 and girth 6 are provided to test the girth variance of the obtained irregular QC codes. Simulations verify the construction of the QC LDPC codes to be valid, since better BER performance than that of existing encodable irregular LDPC codes is obtained.
Since the weight and distance computations of LDPC codes is a NP problem, most presented evaluations of LDPC codes are based on the girth numbers of the codes, and the computation for the distance of the codes is also depended on the girth numbers. There has been no simple way to obtain the distance of LDPC codes because of the code length. Based on generation matrices, this paper develops an algorithm to estimate the upper bound of minimum weights and minimum distance of LDPC codes. The paper establishes the upper bounds of minimum weights and distances of LDPC codes by using the vectors of generation matrices, which is different from the presented probabilitistic search methods for the weights' and distances of LCPC codes. The proposed algorithm can greatly reduce the searching time of weights and distances. Applying the algorithm we find that some LDPC codes get good BER performance although they are only free girth 4 but of great distances. Simulations verify the algorithms to be valid.
To find the minimum distance computation of LDPC codes is a NP problem, there has been no simple way to obtain the minimum distance of LDPC codes because of the code length being very long. To provide a solution to deal with this hard problem, this paper develops an algorithm to estimate the weights and distance of LDPC codes based on generator matrices. The paper establishes the upper bounds of weights and distances of LDPC codes by using the vectors of generator matrices, which is different from the existed probabilistic search methods for the weights' and distances of LCPC codes. The proposed algorithm can greatly reduce the searching time of weights and distances. Applying the algorithm we can obtain some LDPC codes of great distances and being free girth 4. Simulations verify the algorithms to be valid.
The existing design of quasi-cyclic(QC) low-density parity-check(LDPC) codes has not considered the problems of short length girths and the rows' dependency.The first problem leads to the BER performance of QC LDPC codes to be much poorer than that of randomly constructed LDPC codes,while the second problem leads to the difficulty of construction of generator matrices from parity-check matrices.To solve the first problem,the constraint conditions for designing the QC LDPC codes are proposed,and the dimension and shift factors of the circulant matrices of the given sparse parity-check matrix are adjusted according to the test results of both girth 4 and girth 6.To solve the second problem,an approach of irregular QC LDPC codes is proposed.The proposed approach is to replace some sub-matrices by zero matrices and identity matrices at special positions in the given sparse parity-check matrix so as to get a nonsingular square matrix for the construction of the generator matrix.Though adopting bidiagonal submatrices in parity-check matrix can solve the rows' dependency problem,many code words with low weights will occur,which leads to the BER performance can not be better by increasing code lengths.Examples are provided for the proposed design of QC LDPC codes,and computer simulation results show that the proposed QC LDPC codes achieve good BER performance.
The existing construction of Quasi-Cyclic low-density parity-check (short for QC-LDPC) codes has not considered the problems of small stopping sets and small girth and small minimum code weight together, while their existences will lead to the BER performance of QC-LDPC codes to be much poorer than that of randomly constructed LDPC codes even decoding failure. To solve the problem, some theorems of the QC-LDPC codes without small stopping sets and small girth were proposed first. Then good QC-LDPC codes without small stopping sets and small girth were designed based on the row vectors of generator matrix. The small minimum code weight also can be prevented due to the new algorithm based on the row vectors of generator matrix, which is the new way to choose the shifted factors to construct the good QC-LDPC codes for the given prime. The effectiveness and the practicability of the algorithm are demonstrated by the simulation results. It is significant for us to analyze and optimize the design of LDPC codes.
In this paper we present a novel two step design technique for Low Density Parity Check (LDPC) codes, which, among the others, have been exploited for performance enhancement of the second generation of Digital Video Broadcasting- Satellite (DVB-S2). In the first step we develop an efficient algorithm for construction of quasi-random LDPC codes via minimization of a cost function related to the distribution of the length of cycles in the Tanner graph of the code. The cost function aims at constructing high girth bipartite graphs with reduced number of cycles of low length. In the second optimization step we aim at improving the asymptotic performance of the code via edge perturbation. The design philosophy is to avoid asymptotically weak LDPCs that have low minimum distance values and could potentially perform badly under iterative soft decoding at moderate to high Signal to Noise Ratio (SNR) values. Subsequently, we present sample results of our LDPC design strategy, present their simulated performance over an AWGN channel and make comparisons to some of the construction methods presented in the literature