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Journal of Electrical & Electronic Education

A Method for the Estimation of the Minimum Distance of LDPC Codes

Piming Ma

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

LDPC codes provide a lot of advantages,such as low complexity and excellent error-correcting performance.Thus they have been adopted as a channel coding scheme in OFDMA PHY layer of WiMAX802.16e standard.In this paper,we use the minimum distance and code weight distribution to study the performance of LDPC codes.The ANC algorithm is studied deeply and is used to calculate the minimum distance of LDPC codes.Estimation results verify the correctness of ANC algorithm,and also show the influence of the minimum distance and code weight distribution on the error-correcting perfor-mance of LDPC codes.The results provide a reference for selecting LDPC codes in practical design.

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LDPC codes provide a lot of advantages,such as low complexity and excellent error-correcting performance.Thus they have been adopted as a channel coding scheme in OFDMA PHY layer of WiMAX802.16e standard.In this paper,we use the minimum distance and code weight distribution to study the performance of LDPC codes.The ANC algorithm is studied deeply and is used to calculate the minimum distance of LDPC codes.Estimation results verify the correctness of ANC algorithm,and also show the influence of the minimum distance and code weight distribution on the error-correcting perfor-mance of LDPC codes.The results provide a reference for selecting LDPC codes in practical design.
Approximately nearest codeword (ANC) is a classic approach for detecting minimum distance of low-density parity-check (LDPC) codes. With the principle of local search, ANC method is capable of detecting minimum distance of LDPC codes within acceptable time. However, the message passing (MP) decoding algorithm and scheduling scheme, as well as the local search strategy, still have much space to improve. To fill the gap, this paper proposes an improved-ANC method which employs novel noise introduction method, decoding algorithm, scheduling scheme and searching strategy, to accelerate the computation of the minimum distance of LDPC codes significantly. Finally, we run simulations on LDPC codes in Wi-Fi5 and 5G-NR standards. The simulation results are presented and demonstrate the effectiveness of our proposal.
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.
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.
Abstract The evaluation of the minimum distance of low‐density parity‐check (LDPC) codes remains an open problem due to the rather large dimension of the parity check matrix H associated with any practical code. In this article, we propose an effective modification of the error impulse (EI) technique for computation of the minimum distance of the LDPC codes. The EI method is successfully applied to sub‐optimum decoding algorithms such as the iterative MAP decoding algorithm for turbo codes. We present novel modifications and extensions of this method to the sub‐optimum iterative sum‐product algorithm for LDPC codes. The performance of LDPC codes may be limited by pseudo‐codewords. There are, however, cases when the LDPC decoder behaves as a maximum‐likelihood (ML) decoder. This is specially so for randomly constructed LDPC codes operating at medium to high SNR values. In such cases, estimation of the minimum distance d m using the error‐impulse method can be useful to assess asymptotic performance. In short, apart from theoretical interest in achievable d m , the technique is useful in checking whether an LDPC code is poor by virtue of having a low d m . But, if a code has a high d m , simulations would still be needed to assess real performance. Copyright © 2006 AEIT.
The evaluation of the minimum distance of low-density parity-check (LDPC) codes remains an open problem due to the rather large dimension of the parity check matrix H associated with any practical code. In this article, we propose an effective modification of the error impulse (EI) technique for estimation of the minimum distance of the LDPCs. The EI method is successfully applied to suboptimum decoding algorithms such as the iterative MAP decoding algorithm for turbo codes. We present novel modifications and extensions of this method to the suboptimum iterative sum-product algorithm for LDPCs. Simulation results validate the functionality of the proposed technique. Simulations focus on a particular class of LDPC codes, but our approach is general and applies to any LDPC code.
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.