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Quasi-cyclic LDPC codes with high girth  [PDF]
Christian Spagnol,Marta Rossi,Massimiliano Sala
Mathematics , 2009,
Abstract: We study a class of quasi-cyclic LDPC codes. We provide precise conditions guaranteeing high girth in their Tanner graph. Experimentally, the codes we propose perform no worse than random LDPC codes with their same parameters, which is a significant achievement for algebraic codes.
Hierarchical and High-Girth QC LDPC Codes  [PDF]
Yige Wang,Stark C. Draper,Jonathan S. Yedidia
Mathematics , 2011,
Abstract: We present a general approach to designing capacity-approaching high-girth low-density parity-check (LDPC) codes that are friendly to hardware implementation. Our methodology starts by defining a new class of "hierarchical" quasi-cyclic (HQC) LDPC codes that generalizes the structure of quasi-cyclic (QC) LDPC codes. Whereas the parity check matrices of QC LDPC codes are composed of circulant sub-matrices, those of HQC LDPC codes are composed of a hierarchy of circulant sub-matrices that are in turn constructed from circulant sub-matrices, and so on, through some number of levels. We show how to map any class of codes defined using a protograph into a family of HQC LDPC codes. Next, we present a girth-maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high-girth HQC LDPC code. Finally, we discuss how certain characteristics of a code protograph will lead to inevitable short cycles, and show that these short cycles can be eliminated using a "squashing" procedure that results in a high-girth QC LDPC code, although not a hierarchical one. We illustrate our approach with designed examples of girth-10 QC LDPC codes obtained from protographs of one-sided spatially-coupled codes.
Deterministic Constructions for Large Girth Protograph LDPC Codes  [PDF]
Asit Kumar Pradhan,Arunkumar Subramanian,Andrew Thangaraj
Mathematics , 2013,
Abstract: The bit-error threshold of the standard ensemble of Low Density Parity Check (LDPC) codes is known to be close to capacity, if there is a non-zero fraction of degree-two bit nodes. However, the degree-two bit nodes preclude the possibility of a block-error threshold. Interestingly, LDPC codes constructed using protographs allow the possibility of having both degree-two bit nodes and a block-error threshold. In this paper, we analyze density evolution for protograph LDPC codes over the binary erasure channel and show that their bit-error probability decreases double exponentially with the number of iterations when the erasure probability is below the bit-error threshold and long chain of degree-two variable nodes are avoided in the protograph. We present deterministic constructions of such protograph LDPC codes with girth logarithmic in blocklength, resulting in an exponential fall in bit-error probability below the threshold. We provide optimized protographs, whose block-error thresholds are better than that of the standard ensemble with minimum bit-node degree three. These protograph LDPC codes are theoretically of great interest, and have applications, for instance, in coding with strong secrecy over wiretap channels.
New LDPC Codes Using Permutation Matrices with Higher Girth than QC-LDPC Codes Constructed by Fossorier  [PDF]
Mehdi samadieh,Mohammad Gholami
Computer Science , 2014,
Abstract: In the literatures, it is well-known that Fossorier code has the girth among LDPC codes. In this paper, we introduce a new class of low-density parity-check (LDPC) codes, with higher girth than other previous constructed codes. Especially we proposed a new method to construct LDPC codes using non ?xed shift permutation matrices and full based matrices with higher girth than codes constructed by Fossorier.
Searching for Voltage Graph-Based LDPC Tailbiting Codes with Large Girth  [PDF]
Irina E. Bocharova,Florian Hug,Rolf Johannesson,Boris D. Kudryashov,Roman V. Satyukov
Mathematics , 2011, DOI: 10.1109/TIT.2011.2176717
Abstract: The relation between parity-check matrices of quasi-cyclic (QC) low-density parity-check (LDPC) codes and biadjacency matrices of bipartite graphs supports searching for powerful LDPC block codes. Using the principle of tailbiting, compact representations of bipartite graphs based on convolutional codes can be found. Bounds on the girth and the minimum distance of LDPC block codes constructed in such a way are discussed. Algorithms for searching iteratively for LDPC block codes with large girth and for determining their minimum distance are presented. Constructions based on all-ones matrices, Steiner Triple Systems, and QC block codes are introduced. Finally, new QC regular LDPC block codes with girth up to 24 are given.
Construction of High Girth and Two Column Weight LDPC Code Based on Graph  [PDF]
Mahnaz Ahmadi,Hadi Dehghani,Saeid Alikhani,Roslan Hasni
Journal of Applied Sciences , 2012,
Abstract: We present a method for construction of LDPC codes with 2-column weight. This method is based on graph. By considering a LDPC code, we can construct a 2-column weight LDPC code. In this method, we obtain a parity check matrix mxn from a parity check matrix with m rows and n columns for a LDPC code. The girth g in initial check matrix increases to 2g in new check matrix. This increasing is effective in decoding of code. From the simulation result, we observe that the new LDPC code has better performance than Mackay and nil codes in the field Fq.
Girth-12 Quasi-Cyclic LDPC Codes with Consecutive Lengths  [PDF]
Guohua Zhang,Xinmei Wang
Mathematics , 2010,
Abstract: 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.
Tight lower bound of consecutive lengths for QC-LDPC codes with girth twelve  [PDF]
Zhang GuoHua,Wang XinMei
Mathematics , 2012,
Abstract: For an arbitrary (3,L) QC-LDPC code with a girth of twelve, a tight lower bound of the consecutive lengths is proposed. For an arbitrary length above the bound the resultant code necessarily has a girth of twelve, and for the length meeting the bound, the corresponding code inevitably has a girth smaller than twelve. The conclusion can play an important role in the proofs of the existence of large-girth QC-LDPC codes, the construction of large-girth QC-LDPC codes based on the Chinese remainder theorem, and the construction of LDPC codes with the guaranteed error correction capability.
Design of Structured LDPC Codes with Large Girth

ZHANG Wei,ZHU Guang-xi,PENG Li,SHEN Qiong-xia,

计算机科学 , 2009,
Abstract: A parity-check matrix H with large girth has important significance to improve the performance of LDPC codes. And the key to the encoder implementation is the algebraic code structure. This paper proposed a novel code construction algorithm with low complexity based on the Column-Difference Scarch(CDS) Algorithm, which can design regular Quasi Cyclic LDPC codes with large girth and arbitrary code rate. It has linear encoding complexity and is friendly to hardware implementation. The experimental results show that CDS-LDPC codes with different code rates perform better than Tanner codes and Array codes,which increase 0. 79--3. 28dB than another two classical QC-LDPC codes,and also outperform the counterparts of random codes. In addition,CDS-LDPC codes have more flexibility on the design of code length and rate.
A Method for Designing Quasi-Cyclic LDPC Codes Based on Girth Optimization

Xu Hua Xu Cheng-qi,

电子与信息学报 , 2008,
Abstract: The key to improving the performance of QC LDPC codes is how to construct a parity-check matrix H with a girth distribution as good as possible. In this paper, a novel algorithm for constructing QC LDPC codes, GirthOpt-DE algorithm, is proposed, which achieves a good girth distribution based on the differential evolution. Simulation results show that the performance of the QC LDPC codes constructed with the proposed algorithm is superior to Array codes and Tanner codes in both BER and the minimum distance. Besides, the proposed algorithm is more flexible for designing the QC LDPC codes with desired block length and rate as well as good girth.
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