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Evaluations of Good LDPC Codes Based on Generator Matrices

Yang Xiao

Published Jan 1, 2006• 6 citations
Publisher

Abstract

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.

DOI
10.1109/icosp.2006.345907

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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 expurgated bound is proposed to evaluate good LDPC codes in this paper. To get the expurgated bound for LDPC codes needs to know the weights' distribution of LDPC codes to be evaluated, while the calculation of the weights' distribution is NP problem. To solve the problem, an expurgated bound of LDPC codes based on generator matrices is developed. Since the weights' distribution is obtained by the tested generator matrix of given LDPC codes, the expurgated bound of the given codes can reflect the BER performance of the codes. Simulations verify the results to be valid
Finding the minimum distance of low-density-parity-check (LDPC) codes is an NP-hard problem. Different from all existing works that focus on binary LDPC codes, we in this paper aim to compute the minimum distance of nonbinary LDPC codes, motivated by the fact that operating in a large Galois field provides one important degree of freedom to achieve both good waterfall and error-floor performance. Our method is based on the existing nearest nonzero codeword search (NNCS) method, but several modifications are incorporated for nonbinary LDPC codes, including the modified error impulse pattern, the dithering method, and the nonbinary decoder. Numerical results on the estimated minimum distances show that a code's minimum distance can be increased by careful selection of nonzero elements of the parity check matrix, or by increasing the mean column weight, or by increasing the size of the Galois field. These results support observations that have been made based on simulated performance in the literature. Finally, we provide an upper bound on the minimum distance for nonbinary quasi-cyclic LDPC codes.
We investigate the minimum distance of structured binary Low-Density Parity-Check (LDPC) codes whose parity-check matrices are of the form [C∣M][\mathbf{C} \vert \mathbf{M}][C∣M] where C\mathbf{C}C is circulant and of column weight 222, and M\mathbf{M}M has fixed column weight r≥3r \geq 3r≥3 and row weight at least 111. These codes are of interest because they are LDPC codes which come with a natural linear-time encoding algorithm. We show that the minimum distance of these codes is in O(nr−2r−1+ε)O(n^{\frac{r-2}{r-1} + ε})O(nr−1r−2​+ε), where nnn is the code length and ε>0ε> 0ε>0 is arbitrarily small. This improves the previously known upper bound in O(nr−1r)O(n^{\frac{r-1}{r}})O(nrr−1​) on the minimum distance of such codes.
In this letter, we obtain lower and upper bounds on the minimum distance dminof low-density parity-check (LDPC) codes. The bounds are derived by categorizing the non-zero code words of an LDPC code into two categories of elementary and non-elementary. The first category contains code words whose induced subgraph has only degree-2 check nodes. We propose an efficient search algorithm that can find the elementary code words of an LDPC code with weight less than a certain value amax, exhaustively. We also derive a lower bound Lneon the weight of non-elementary code words. By performing the search with amax= Lne, we either obtain an elementary code word with the smallest weight dmin, or establish the lower bound of Lneon dmin. For the upper bound, we modify our search algorithm to reach elementary codewords of larger weights at the cost of being non-exhaustive. Once such a codeword is found, its weight acts as an upper bound on dmin. We examine a large number of regular and irregular LDPC codes, and demonstrate the efficiency and versatility of our technique in finding lower and upper bounds on, and in many cases the exact value of, dmin. Finding dmin, or establishing search-based lower or upper bounds, for many of the examined codes are out of the reach of any existing algorithm.
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.