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IEEE Communications Letters

QC-LDPC Codes with Girth Eight Based on Independent Row-Column Mapping Sequence

Lei Wang, Xing Zhang, Feng Yu, Yao-Xian Fu, Yingguan Wang

Published Oct 10, 2013• 6 citations
Publisher

Abstract

Quasi-cyclic low-density parity-check (QC-LDPC) codes are known to have algebraic structure, which suits hardware implementation. However, low girth makes their performances worse than random structured LDPC codes. Based on independent row-column mapping sequence, this work proposed a generalized method of constructing QC-LDPC codes with girth eight, which enlarges the class of girth eight QC-LDPC codes to a fairly general series. Simulation results show that the bit error rate of the constructed codes is close to that of progressive edge growth (PEG) based QC-LDPC codes under iterative decoding.

DOI
10.1109/lcomm.2013.092813.131581

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Quasi-cyclic low-density parity-check (QC-LDPC) codes are known to have algebraic structure, which suits hardware implementation. However, low girth makes their performances worse than random structured LDPC codes. Based on independent row-column mapping sequence, this work proposed a generalized method of constructing QC-LDPC codes with girth eight, which enlarges the class of girth eight QC-LDPC codes to a fairly general series. Simulation results show that the bit error rate of the constructed codes is close to that of progressive edge growth (PEG) based QC-LDPC codes under iterative decoding.
In this paper, we consider the constructions of type-II quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight from a Sidon sequence. We first derive the necessary and sufficient conditions guaranteeing a girth-eight type-II QC-LDPC code. By combining these conditions with the concept of the Sidon sequence, three classes of type-II QC-LDPC codes are subsequently proposed with girth eight. To the best of our knowledge, the second and the third classes we proposed are the first series of systematic and algebraic constructions for the girth-eight type-II QC-LDPC codes that have rates being greater than a half and, at the same time, have distance upper bounds being not limited to 12. Via simulations, we show the promising performance of the proposed type-II QC-LDPC codes with girth eight. In addition, a set of general bounds and explicit/random constructions without using a Sidon sequence are also presented for the type-II QC-LDPC codes with girth six or eight. We observe that the proposed codes perform better than or equally well as the random QC-LDPC codes from PEG and quadr.-congr. methods, while the novel codes possess both highly structured parity-check matrices and very flexible choices in circulant size.
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 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.
Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.
Based on cycle search algorithm and Progressive Edge Growth(PEG) algorithm,this paper proposes a method to construct quasi-cyclic Low-density Parity Check(LDPC) codes with big girth and linear-encoding complexity.A parity-check matrix with a form of approximate lower triangle structure is designed to achieve iterative encoding with linear-encoding complexity.PEG algorithm is used to construct the base matrix and the cycle search algorithm is used to optimize the cycle distribution.And a fast encoding method is proposed.Experimental results show that the LDPC codes constructed with the proposed method can avoid error floor and achieve better BER performance than Tanner-QC-LDPC codes and equivalent BER performance to the ones of random codes.