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Eigenvalue bounds on the pseudocodeword weight of expander codes [article]

Christine A. Kelley, Deepak Sridhara
2007 arXiv   pre-print
For each case, lower bounds on the minimum stopping set size and the minimum pseudocodeword weight of expander (LDPC) codes are derived.  ...  These bounds are compared with the known eigenvalue-based lower bounds on the minimum distance of expander codes.  ...  We also thank Reviewer 1 for providing the more intuitive definition of a stopping set in a generalized LDPC code.  ... 
arXiv:0708.2462v1 fatcat:5r7thuq535cjxnzrjwznougqpe

Eigenvalue bounds on the pseudocodeword weight of expander codes

Deepak Sridhara, Christine Kelley
2007 Advances in Mathematics of Communications  
For each case, lower bounds on the minimum stopping set size and the minimum pseudocodeword weight of expander (LDPC) codes are derived.  ...  These bounds are compared with the known eigenvalue-based lower bounds on the minimum distance of expander codes.  ...  We also thank Reviewer 1 for providing the more intuitive definition of a stopping set in a generalized LDPC code.  ... 
doi:10.3934/amc.2007.1.287 fatcat:rwnfklidyffblm5uoipekeml6y

Coding Theory

Joachim Rosenthal, Mohammad Amin Shokrollahi
2007 Oberwolfach Reports  
The workshop on coding theory covered many facets of the recent research advances.  ...  Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete  ...  Eigenvalue bounds on the pseudocodeword weight of expander codes Deepak Sridhara (joint work with Christine A.  ... 
doi:10.4171/owr/2007/56 fatcat:ac2rttsrn5g2rn5la72nudiyum

Impact of redundant checks on the LP decoding thresholds of LDPC codes [article]

Louay Bazzi, Hani Audah
2015 arXiv   pre-print
We prove that for LDPC codes, even if we include all redundant checks, asymptotically there is no gain in the LP decoder threshold on the BSC under certain conditions on the base Tanner graph.  ...  We call the graph rigid if the minimum weight of a sum of check nodes involving a cycle tends to infinity as the block length tends to infinity.  ...  Acknowledgments The authors would like to thank the anonymous reviewers for their helpful and constructive comments.  ... 
arXiv:1411.7554v3 fatcat:ts7cyamgfzbivevenzhmcm5kj4

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

Christine A. Kelley, Deepak Sridhara, Joachim Rosenthal
2007 arXiv   pre-print
The performance of codes defined from graphs depends on the expansion property of the underlying graph in a crucial way.  ...  The paper investigates the use of zig-zag and replacement product graphs for the construction of codes on graphs.  ...  Furthermore, graphs with good expansion yield LDPC codes with good minimum distance and pseudocodeword weights [4, 6, 7, 14, 15] .  ... 
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 performance of codes defined from graphs depends on the expansion property of the underlying graph in a crucial way.  ...  The paper investigates the use of zig-zag and replacement product graphs for the construction of codes on graphs [16] .  ...  Furthermore, graphs with good expansion yield LDPC codes with good minimum distance and pseudocodeword weights [4, 6, 7, 14, 15] .  ... 
doi:10.3934/amc.2008.2.347 fatcat:k3isnbeki5hsjpiiwaxt2dqaia

Correcting a Fraction of Errors in Nonbinary Expander Codes With Linear Programming

Vitaly Skachek
2011 IEEE Transactions on Information Theory  
It is also shown that this decoder corrects any pattern of errors of a relative weight up to approximately 1/4 δ_A δ_B (where δ_A and δ_B are the relative minimum distances of the constituent codes).  ...  A linear-programming decoder for nonbinary expander codes is presented. It is shown that the proposed decoder has the maximum-likelihood certificate properties.  ...  ACKNOWLEDGMENTS The author wishes to thank Marcus Greferath.  ... 
doi:10.1109/tit.2011.2137030 fatcat:hkqgw3n5bvavxk2vyyjzmqrd4a

Efficient implementation of linear programming decoding [article]

Mohammad H. Taghavi, Amin Shokrollahi, Paul H. Siegel
2009 arXiv   pre-print
at most one linear constraint derived from each of the parity-check constraints.  ...  This paper explores ideas for fast LP decoding of low-density parity-check (LDPC) codes.  ...  An example is the pseudocodeword [1, 1 2 , 0, 1 2 , 0, 0, 1 2 ] of the (7, 4, 3) code with m = 3, given in [4] . C.  ... 
arXiv:0902.0657v1 fatcat:fzdmaxpoc5haxglzmv7quc7vqq

Efficient Implementation of Linear Programming Decoding

Mohammad H. Taghavi, Amin Shokrollahi, Paul H. Siegel
2011 IEEE Transactions on Information Theory  
We first propose a modification of Adaptive LP decoding, and prove that it performs LP decoding by solving a number of linear programs that contain at most one linear constraint derived from each of the  ...  Then, as a step toward designing an efficient LP solver that exploits the structure of LDPC codes, we study a sparse interior-point implementation for solving this sequence of linear programs.  ...  The distribution of the eigenvalues of the coefficient matrix Q has a crucial effect on the convergence behavior of the CG method.  ... 
doi:10.1109/tit.2011.2161920 fatcat:5ny6bl2ybvcondyiev3mlnxefa

Efficient implementation of linear programming decoding

Mohammad H. Taghavi, Amin Shokrollahi, Paul H. Siegel
2008 2008 46th Annual Allerton Conference on Communication, Control, and Computing  
We first propose a modification of Adaptive LP decoding, and prove that it performs LP decoding by solving a number of linear programs that contain at most one linear constraint derived from each of the  ...  Then, as a step toward designing an efficient LP solver that exploits the structure of LDPC codes, we study a sparse interior-point implementation for solving this sequence of linear programs.  ...  The distribution of the eigenvalues of the coefficient matrix Q has a crucial effect on the convergence behavior of the CG method.  ... 
doi:10.1109/allerton.2008.4797586 fatcat:rx3ulujkyjcqfc6lzakyquopeu

Graphical Models, Exponential Families, and Variational Inference

Martin J. Wainwright, Michael I. Jordan
2007 Foundations and Trends® in Machine Learning  
of the problems of computing likelihoods, marginal probabilities and most probable configurations.  ...  Many problems that arise in specific instancesincluding the key problems of computing marginals and modes of probability distributions -are best studied in the general setting.  ...  The intellectual contributions and support of Alan Willsky and Tommi Jaakkola were particularly significant in the development of the ideas presented here.  ... 
doi:10.1561/2200000001 fatcat:3f33bwasgvg5ndjfqezocaxxfa

Controlling the Error Floors of the Low-Density Parity-Check Codes

Shuai Zhang
2013
However, by Theorem 2.3 the weight of instantons is lower bounded by d frac /2 .  ...  If M (f ) is decoded to some pseudocodeword f , it can be shown that the weight of this f must be no more than the weight of f [11] . Step 3.  ...  In addition, the dominant eigenvalue of the VC matrix is 4 and the corresponding eigenvector is an all-1 vector.  ... 
doi:10.7939/r39p42 fatcat:mjdjfh3fyrbffkb7gvajo3gnny

Analysis of Dynamic Social and Technological Networks (Dagstuhl Seminar 11452) Coding Theory (Dagstuhl Seminar 11461)

Holger Giese, Bernhard Rumpe, Bernhard Schätz, Janos Sztipanovits, Lars Elden, Andreas, Vito Latora, Cecilia Mascolo, Mirco, Joachim Rosenthal, M Shokrollahi, Judy
2011 Science and Engineering of Cyber-Physical Systems   unpublished
We recently proposed a formulation for the Requirements Problem of SAS (revisiting foundational work by Zave et. al., TOSEM'97, and a more recent one by Jureta et al., RE'08), as a dynamic problem (Qureshi  ...  In our approach the requirements problem is represented through a goal-oriented model, which needs to be dynamically updated in order to reflect run-time changes in the requirements problem.  ...  Also, we provide upper bounds on the pseudocodeword redundancy for some families of codes with t-transitive automorphism group, and we report on our progress of proving bounds for the extended Golay code  ... 
fatcat:bwnjnbq2bjbolk52twaysozyxu

SPARSE GRAPH-BASED CODING SCHEMES FOR CONTINUOUS PHASE MODULATIONS Directeur(s) de Thèse : Rapporteurs : Membre(s) du jury

Mathématiques, M Charly, Poulliat Mme, Marie Boucheret, M Jego, Universite De Bordeaux, M Declercq, M Duhamel, Président Abdel, Nour Charbel
unpublished
) = λ 2 , it gives an upper bound on the proportion λ 2 .  ...  Expanding this inequality concludes the proof.  ...  compute the limit of the left function in Eq.  ... 
fatcat:i5nrf5k3kbe2joli3qbpikjjaa