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Zig-zag and Replacement Product Graphs and LDPC Codes [article]

Christine A. Kelley, Deepak Sridhara, Joachim Rosenthal
2007 arXiv   pre-print
The paper investigates the use of zig-zag and replacement product graphs for the construction of codes on graphs.  ...  Graph products, such as the zig-zag product and replacement product provide new infinite families of constant degree expander graphs.  ...  Zig-zag and replacement product LDPC codes In this section, we design LDPC codes based on expander graphs arising from the zig-zag and replacement products.  ... 
arXiv:cs/0611155v2 fatcat:3ex3qwbknnhltmvod65dbbtrai

Zig-zag and replacement product graphs and LDPC codes

Christine Kelley, Deepak Sridhara, Joachim Rosenthal
2008 Advances in Mathematics of Communications  
The paper investigates the use of zig-zag and replacement product graphs for the construction of codes on graphs [16] .  ...  Graph products, such as the zig-zag product [13] and replacement product provide new infinite families of constant degree expander graphs.  ...  Zig-zag and replacement product LDPC codes In this section, we design LDPC codes based on expander graphs arising from the zig-zag and replacement products.  ... 
doi:10.3934/amc.2008.2.347 fatcat:k3isnbeki5hsjpiiwaxt2dqaia

Randomness conductors and constant-degree lossless expanders

Michael Capalbo, Omer Reingold, Salil Vadhan, Avi Wigderson
2002 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing - STOC '02  
In these graphs, the expansion factor is almost as large as possible: (1 − ǫ)D, where D is the degree and ǫ is an arbitrarily small constant.  ...  The main abstract contribution of this paper is the introduction and initial study of randomness conductors, a notion which generalizes extractors, expanders, condensers and other similar objects.  ...  discussions about the applications to codes, and the referees for helpful comments on the presentation.  ... 
doi:10.1145/510002.510003 fatcat:zrdsw422zbdtpowpnj6a4lgs3e

Randomness conductors and constant-degree lossless expanders

Michael Capalbo, Omer Reingold, Salil Vadhan, Avi Wigderson
2002 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing - STOC '02  
In these graphs, the expansion factor is almost as large as possible: (1 − ǫ)D, where D is the degree and ǫ is an arbitrarily small constant.  ...  The main abstract contribution of this paper is the introduction and initial study of randomness conductors, a notion which generalizes extractors, expanders, condensers and other similar objects.  ...  discussions about the applications to codes, and the referees for helpful comments on the presentation.  ... 
doi:10.1145/509907.510003 dblp:conf/stoc/CapalboRVW02 fatcat:aowwozdzvvapdji2f6ycmut5gq

LDPC Codes in Communications and Broadcasting

T. OHTSUKI
2007 IEICE transactions on communications  
Moreover, we present some research on LDPC coded MIMO systems and HARQ using LDPC codes.  ...  We also overview some standardized LDPC codes, the LDPC codes standardized in DVB-S2 and the IEEE802.16e standard LDPC codes.  ...  The connections between check nodes and information bit nodes are determined randomly, while those between check nodes and parity bit nodes are arranged in a regular zig-zag pattern so that the encoding  ... 
doi:10.1093/ietcom/e90-b.3.440 fatcat:umyh6rrasre37f77cl6zqhbnge

Design of structured eIRA codes with applications to wireless channels

Yifei Zhang, W.E. Ryan, Fei Peng
2006 IEEE Wireless Communications and Networking Conference, 2006. WCNC 2006.  
These SeIRA codes permit flexibility in code dimension, length, and rate, while facilitating encoder/decoder implementation due to their structure.  ...  We present a design algorithm for SeIRA codes and software-based performance results which demonstrate their low floors.  ...  It is also mentioned in [12] for IRA codes that zig-zag pattern of H 2 permits a very simple puncturing pattern: the parity bits are punctured in an alternating pattern.  ... 
doi:10.1109/wcnc.2006.1696490 dblp:conf/wcnc/ZhangRP06 fatcat:gyfmgiu7yvauxahjyltyz5piqe

Expander graphs and their applications

Shlomo Hoory, Nathan Linial, Avi Wigderson
2006 Bulletin of the American Mathematical Society  
What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness.  ...  Expansion of a graph requires that it is simultaneously sparse and highly connected.  ...  He is mostly interested in the interplay between mathematics, computer science, and other fields of science. Avi Wigderson is a professor at the Institute for Advanced Study in Princeton.  ... 
doi:10.1090/s0273-0979-06-01126-8 fatcat:5u65bvautndzhnkhxurqrudqfy

Interleave-division multiple access (IDMA) using low-rate layered LDGM codes

Francisco J. Vázquez-Araújo, Miguel González-López, Luis Castedo, Javier Garcia-Frias
2010 Wireless Communications and Mobile Computing  
Simulation results show that the designed codes approach the theoretical limits and outperform previous IDMA schemes based on Turbo-Hadamard codes.  ...  We study the behavior of the soft interference cancellation (SIC) detector employed in IDMA systems and design the channel codes using EXtrinsic Information Transfer (EXIT) evolution, aiming at optimizing  ...  01, and CSD2008-00010.  ... 
doi:10.1002/wcm.1055 fatcat:mlrfetgk4ndaxbqyebnlk6j2nq

Explicit Abelian Lifts and Quantum LDPC Codes [article]

Fernando Granha Jeronimo, Tushant Mittal, Ryan O'Donnell, Pedro Paredes, Madhur Tulsiani
2021 arXiv   pre-print
As corollaries, we obtain explicit quantum lifted product codes of Panteleev and Kalachev of almost linear distance (and also in a wide range of parameters) and explicit classical quasi-cyclic LDPC codes  ...  For an abelian group H acting on the set [ℓ], an (H,ℓ)-lift of a graph G_0 is a graph obtained by replacing each vertex by ℓ copies, and each edge by a matching corresponding to the action of an element  ...  These range from very explicit algebraic constructions of socalled Ramanujan graphs [LPS88] to recursive combinatorial ones based on the Zig-Zag product [RVW00] .  ... 
arXiv:2112.01647v1 fatcat:4wevb2ggnnaspknxyuvrauvnke

Good quantum LDPC codes with linear time decoder from lossless expanders [article]

Ting-Chun Lin, Min-Hsiu Hsieh
2022 arXiv   pre-print
We study qLDPC codes constructed from balanced products and lossless expanders.  ...  We found that assuming the existence of 2-sided lossless expander graphs with free group action, the resulting qLDPC codes have constant rate, linear distance, and linear time decoders.  ...  It is known that 1-sided lossless expanders with symmetry exist [15] , so one may hope to generalize their method and find a new version of zig-zag products that gives 2-sided lossless expanders.  ... 
arXiv:2203.03581v1 fatcat:hev2nt7pinendivnbcbw7r3twy

Asymptotic distance and convergence analysis of braided protograph convolutional codes

Marcos B.S. Tavares, Michael Lentmaier, Gerhard P. Fettweis, Kamil Sh. Zigangirov
2008 2008 46th Annual Allerton Conference on Communication, Control, and Computing  
Interestingly, we can also observe that for the irregular ensembles in Fig. 4(b) , the curves of the distance ratios show a zig-zag behavior as if there were different ensembles for odd L and for even  ...  The presented distance bounding technique generalizes some earlier results on permutation-based regular LDPC convolutional code ensembles to the case of irregular base code graphs, and it can be applied  ... 
doi:10.1109/allerton.2008.4797678 fatcat:dd54n2qplnaonayvl2kgmom4lm

Expander Graphs in Pure and Applied Mathematics [article]

Alexander Lubotzky
2011 arXiv   pre-print
They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more.  ...  Expander graphs are highly connected sparse finite graphs.  ...  Thanks are also due to the ERC and ISF for partial support.  ... 
arXiv:1105.2389v1 fatcat:ci3j6nleefge3nqh3tqiihlxca

Optimal quantum subsystem codes in 2-dimensions [article]

Theodore J. Yoder
2019 arXiv   pre-print
Given any two classical codes with parameters [n_1,k,d_1] and [n_2,k,d_2], we show how to construct a quantum subsystem code in 2-dimensions with parameters [[N,K,D]] satisfying N< 2n_1n_2, K=k, and D=  ...  Finally, while the subsystem codes we create do not have asymptotic thresholds, we show how they can be gauge-fixed to certain hypergraph product codes that do.  ...  However, the zig-zag construction [29] eventually solved this problem. For our purposes, a suitable distillation of their result is the following.  ... 
arXiv:1901.06319v2 fatcat:ypo2iu3ehjcjpbmfeydxbyki2u

Distributed Video Coding: CODEC Architecture and Implementation

Vijay Kumar Kodavalla, P.G Krishna Mohan
2011 Signal & Image Processing An International Journal  
Distributed Video Coding (DVC) is a new coding paradigm for video compression, based on Slepian- Wolf (lossless coding) and Wyner-Ziv (lossy coding) information theoretic results.  ...  coding)[4][5] in 2002 and European project DISCOVER (DIStributed COding for Video SERvices)[6] in 2007.  ...  inverse zig-zag scan.  ... 
doi:10.5121/sipij.2011.2111 fatcat:m2tmgcpiwjcnfl3wk7j454pb2u

c^3-Locally Testable Codes from Lossless Expanders [article]

Ting-Chun Lin, Min-Hsiu Hsieh
2022 arXiv   pre-print
In this work, we construct a new LTC family using 1-sided lossless expanders and balanced products.  ...  A locally testable code (LTC) is an error correcting code with a property tester.  ...  However, because we need the graph to have symmetry, we use the explicit construction by [15] , where they apply a generalized zig-zag product on Cayley eigenvalue expander graphs.  ... 
arXiv:2201.11369v2 fatcat:bgxicdtrg5apjgcitmayljbsuq
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