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International Conference on Electronics, Information, and Communication (ICEIC)

A base matrix method to construct column weight 3 quasi-cyclic LDPC codes with high girth

Abhishek Kalsi, Ambar Bajpai, Lunchakorn Wuttisittikulkij, Piya Kovintaewat

Published Jan 1, 2016• 2 citations
Publisher

Abstract

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.

DOI
10.1109/elinfocom.2016.7562999

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IEEE Communications Letters•May 23, 2013•14 citations
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.
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.
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.
This article presents a highly flexible method for constructing high-girth quasi-cyclic low-density parity-check (QC-LDPC) codes. The proposed algorithm constructs a Tanner graph formed by connecting groups of rows (check nodes) and columns (variable nodes) of the constructed code. To obtain a sub-matrix structure, rows and columns are divided into groups of equal sizes. Rows and columns in a group are connected in their numerical (positional) order to obtain a cyclic structure. Connected rows and columns must satisfy the desired minimum cycle length. We present conditions that guarantee desired girths. The proposed algorithm is by far more flexible in constructing a wide range (rates and lengths) of regular and irregular QC-LDPC codes compared to existing methods. The algorithm, which has linear complexity with respect to the number of rows or columns, provides an easy and fast way to construct QC-LDPC codes. Constructed codes show good bit error rate performances.
A method to construct girth-12 (3,L) quasi-cyclic low-density parity-check (QC-LDPC) codes with all lengths larger than a certain given number is proposed, via a given girth-12 code subjected to some constraints. The lengths of these codes can be arbitrary integers of the form LP, provided that P is larger than a tight lower bound determined by the maximal element within the exponent matrix of the given girth-12 code. By applying the method to the case of row-weight six, we obtained a family of girth-12 (3,6) QC-LDPC codes for arbitrary lengths above 2688, which includes 30 member codes with shorter code lengths compared with the shortest girth-12 (3,6) QC-LDPC codes reported by O'Sullivan.
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.
In this paper, some well-structured block designs are presented whose incidence matrices can be considered as the mother matrices of some regular and irregular column-weight three quasi cyclic low-density parity-check codes of girth 20. The class of LDPC codes derived from our construction possesses much more flexibility in terms of code length and rate. Moreover, the length of the regular constructed codes with rate at most 0.5 is at least 4.9 times smaller than the length of the voltage graph LDPC codes with the same rate and girth. Simulation results show that the constructed codes outperform progressive edge growth, repeat accumulate and random LDPC codes.