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IEEE International Conference on Communication Systems (ICCS)

Good encodable irregular quasi-cyclic LDPC codes

Yang Xiao, Kiseon Kim

Published Nov 1, 2008• 6 citations
Publisher

Abstract

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.

DOI
10.1109/iccs.2008.4737391

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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.
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.
There is an important breakthrough in the field of forward error correction techniques, low-density parity-check (LDPC) codes. These codes have potential to approach the Shannon limit with reliable performance on a given channel, known as the channel capacity. Recently, researchers pay much more attention towards LDPC codes because its performance is better than Turbo codes. However, the LDPC code is the best when it is an irregular code and uses with a very large block-length. Although, the short block-length LDPC codes perform not so well, they are easy to implement in various practical applications. Therefore, there is still a demand of new development on the encoding side of LDPC codes for various ranges of code rate and code length so as to capable of using in many applications.This thesis proposes the development of quasi-cyclic (QC) LDPC codes for column weight 3. In the first part of this thesis, we propose a new construction algorithm of QC-LDPC codes for medium to large block-length by combining QC-LDPC codes of small block-length as their component codes, via Chinese remainder theorem (CRT). Such component codes were constructed by permuting each column block sequentially to attain the desire local girth. After combining all component codes to generate an expanded parity-check matrix, the resulting girth is greater than or at least equal to the maximum girth of the component codes. Simulation results show that our proposed construction method of the parity-check matrix significantly outperforms the other well-known existing methods in terms of low error-floor, simple structure, high performance, and can reduce encoding complexity.In addition, this thesis also proposes two new construction methods for QC-LDPC codes, namely a base matrix method and a subtraction based method. A base matrix based method is a simple, less computational complexity method for constructing the exponent matrix (3, K) of girth 8, 10 and 12 of QC-LDPC codes. Another method is a subtraction method, which has a similar exponent matrix as a base matrix method and has a girth at least 8. Results indicate that the LDPC codes constructed from these two methods have flexibility for arbitrary block-column length K and have a similar BER performance if compared to existing methods.
Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes require less hardware resources due to the quasi-cyclic structure of parity-check matrices.In this paper, two improved methods for constructing QC-LDPC codes are proposed.Firstly, in order to make sure the masked LDPC codes have flexible code rate and code length, an improved construction of masking matrix is presented.The proposed regular masking matrix of QC-LDPC codes have the girth at least 6.Then, the irregular codes are constructed by using improved finite field approach and PEG algorithm, which result in large girth and less short cycles.Simulations demonstrate that the proposed QC-LDPC codes have both lower error floor and good waterfall performance.
The existing design of quasi-cyclic low-density parity-check(short for QC-LDPC) codes has not considered the problems of small stopping sets and small cycles together,while their existences will lead to the BER performance of QC-LDPC codes to be much poorer than that of the randomly constructed LDPC codes even decoding failure.To solve the problem,some theorems of the QC-LDPC codes without small stopping sets and small cycles were proposed which provide a theoretical basis for the construction of parity-check matrix by analysis method;meanwhile,good QC-LDPC codes without small stopping sets and small cycles were designed.Since preventing the small stopping sets and small cycles also prevents a low minimum distance,our construction method also leads to QC-LDPC codes with a higher minimum distance.The simulation results show that the QC-LDPC codes can effectively avoid specified stopping sets and small cycles and achieve good coding and decoding property due to the proposed constructions.
This letter presents a necessary and sufficient condition for a class of quasi-cyclic low-density parity-check (QC LDPC) codes without girth four. Girth-four property of a class of QC LDPC codes is investigated. Good QC LDPC codes without girth four can be constructed by selecting proper shifting factors according to the proposed theorems. Examples are provided to verify the theorems. The simulation results show that the QC LDPC codes without girth four achieve a better BER performance compared with that of randomly constructed LDPC codes.