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International Siberian Conference on Control and Communications

A subtraction based method for the construction of quasi-cyclic LDPC codes of girth eight

Ambar Bajpai, Lunchakorn Wuttisittikulkij, Abhishek Kalsi, Piya Kovintavewat

Published May 1, 2016• 0 citations
Publisher

Abstract

This article presents a simple, less computational complexity method for constructing exponent matrix (3, K) having girth at least 8 of quasi-cyclic low-density parity-check (QC-LDPC) codes based on subtraction method. The construction of code deals with the generation of exponent matrix by three formulas. This method is flexible for any block-column length K. The simulations are shown in comparison with some existing appreciable work. The codes with girth 8 are constructed with circulant permutation matrix (CPM) size P ≥ max{a2, r, a2, r,….a2, k} + 1.

DOI
10.1109/sibcon.2016.7491830

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This article presents a simple, less computational complexity method for constructing exponent matrix (3, K) having girth at least 8 of quasi-cyclic low-density parity-check (QC-LDPC) codes based on subtraction method. The construction of code deals with the generation of exponent matrix by three formulas. This method is flexible for any block-column length K. The simulations are shown in comparison with some existing appreciable work. The codes with girth 8 are constructed with circulant permutation matrix (CPM) size P ≥ max{a2, r, a2, r,….a2, k} + 1.
This article presents a simple, less computational complexity method for constructing exponent matrix (3, K) of girth 8, 10 and 12 of quasi-cyclic low-density parity-check codes (QC-LDPC) based on generation of base matrix. The construction of code deals with generation of base matrix by a simple algorithm and element wise element method for girth 8, while only element wise element method for girth 10 and 12. These methods are flexible for block-column length K. The simulations are shown in comparison with some existing appreciable work.
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.
Recently, for each row weight KKK and column-weight JJJ, 3leJ<K3le J< K3leJ<K, several classes of (J,K)(J,K)(J,K) quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth 8 have been constructed explicitly such that their corresponding lower-bounds on the size of circulant permutation matrices (CPMs) have been considered small as possible. In this paper, for J=7J=7J=7, a class of (7,K)(7,K)(7,K) QC-LDPC codes with girth 8 is generated by an explicit method such that the lower-bounds of the constructed codes remarkably are better than the state-of-the-art bound (K−1)(K2+K)+1(K-1)(K^2+K)+1(K−1)(K2+K)+1.
Enabling the parity-check matrices to have a large girth is an efficient method to improve the decoding performance for many binary and non-binary QC-LDPC codes.A new explicit framework to construct QC-LDPC codes with girth at least eight is proposed,without any computer search procedures.The QC-LDPC codes constructed by the framework not only have a girth at least eight,but also possess an advantage of consecutive cyclic permutation matrix(CPM) sizes.The framework includes two steps:first,a parity-check matrix is explicitly constructed with a girth at least eight under an infinite CPM size;second,the tight lower bound of consecutive CPM sizes is precisely calculated directly from the corresponding shift matrix of the parity-check matrix,by employing a new girth property discovered by this paper.
In order to construct girth-8 quasi-cyclic (QC) low-density parity-check (LDPC) codes with low encoding complexity, this paper introduces structural shaping techniques to generate QC-LDPC codes with two different structures. These two structures are QC-LDPC codes constructed with vertical symmetric structure and reverse structure without 4-cycles and 6-cycles. Novel constructions assume that the structures of the parity check matrix (PCM) are symmetric and reverse, respectively, and the greedy exhaustive search algorithm is used to find the exponent matrix with the smallest possible size P of the circulant permutation matrix (CPM). Due to its symmetrical and reverse structural characteristics, the representation of the new codes is more compact than other codes. The simulation results show that the proposed codes have certain competitiveness compared to existing QC-LDPC codes.