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Journal of Communications

Explicit construction of girth-eight QC-LDPC codes and its application in CRT method

Xinmei Wang

Published Jan 1, 2012• 8 citations
OpenAlex

Abstract

For arbitrary code lengths of the form PL(P≥3L2/4+L?1),a new family of(4,L)-regular quasi-cyclic(QC) low-density parity-check(LDPC) codes was proposed explicitly with girth eight.Employing the new code as a compo-nent code in the construction method of Chinese remainder theorem(CRT),a novel class of compound QC-LDPC codes was presented with both girth at least eight and very flexible code lengths.Simulation results show that the new compound codes with rate 1/2 and moderate code lengths perform very well over the additive white Gaussian noise(AWGN) channel.

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Design and performance evaluation of low complexity encoding methods for LDPC codes

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•Jan 1, 2015•0 citations
For arbitrary code lengths of the form PL(P≥3L2/4+L?1),a new family of(4,L)-regular quasi-cyclic(QC) low-density parity-check(LDPC) codes was proposed explicitly with girth eight.Employing the new code as a compo-nent code in the construction method of Chinese remainder theorem(CRT),a novel class of compound QC-LDPC codes was presented with both girth at least eight and very flexible code lengths.Simulation results show that the new compound codes with rate 1/2 and moderate code lengths perform very well over the additive white Gaussian noise(AWGN) channel.
In this letter, we consider two problems associated with quasi-cyclic low-density parity-check (QC-LDPC) codes. The first is how to extend the code length of a QC-LDPC code without reducing the girth. The second is how to design a QCLDPC code with a prescribed girth easily. We deal with these two problems by using a combining method of QC-LDPC codes via the Chinese Remainder Theorem (CRT). Codes constructed with our proposed method have flexible code lengths, flexible code rates and large girth. Simulation results show that they perform very well with the iterative decoding.
In this paper we propose a method of constructing quasi-cyclic low-density parity-check (QC-LDPC) codes of large length by combining QC-LDPC codes of small length as their component codes, via the Chinese remainder theorem. The girth of the QC-LDPC codes obtained by the proposed method is always larger than or equal to that of each component code. By applying the method to array codes, we present a family of high-rate regular QC-LDPC codes with no 4-cycles. Simulation results show that they have almost the same performance as random regular LDPC codes.
An improved combining method for designing quasi- cyclic low- density parity- check( QC-LDPC) codes based on Chinese Remainder Theorem( CRT) is proposed to improve the error- correcting performance by increasing the girth and decreasing the number of short cycles. With the CRT combining method,the difficult problem of designing QC- LDPC codes with girth g is translated into a task of designing one component code with girth g. By properly permuting the column blocks of parity- check matrices of other component codes,a lot of QC- LDPC codes with much shorter cycles and better performance can be designed,which are more suitable for communication systems with request of high reliability. Simulations show that compared with existing CRT- based QC- LDPC codes,the proposed QC- LDPC codes have much shorter cycles and obtain 1. 2 d B coding gain at bit error rate( BER) of 10- 6.
For any column weight J and any row weight L, a novel framework is proposed such that a girth-eight (J,L) quasi-cyclic low-density parity-check (QC-LDPC) code with any block length above a lower bound can be constructed via a simple inequality in terms of greatest common divisor (GCD). The main advantage is that the construction of a class of (J,L) girth-eight QC-LDPC codes is transformed into a rather simple task, searching for J integers satisfying the so-called GCD constraint for L. Combining the new method with masking matrices, a class of type-1 QC-LDPC codes is presented with girth at least eight. Simulation results show that the type-1 codes perform better than the random QC-LDPC codes and quadratic-congruence-based QC-LDPC codes for moderate block lengths and low code rates.
This paper proposes a coset-based Quasi-Cyclic Low Density Parity Check(QC LDPC) construction method and similar SFT-LDPC code construction method in order to guarantee the girth under high bitrate.It gives two necessary conditions of girth at least 6 when the size of cyclic permutation matrix is a composite number.This construction method with various lengths,rates and girth at least 6 can be flexiblely constructed.Simulation results show that the code constructed by this method is comparable to random ones in error correction performance over additive white Gaussian noise channel.
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.