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Pseudocodeword weights for non-binary LDPC codes

Christine Kelley, Deepak Sridhara, Joachim Rosenthal
2006 2006 IEEE International Symposium on Information Theory  
The tree-based lower bounds on the minimum pseudocodeword weight are shown to also hold for q-ary LDPC codes on these channels.  ...  Pseudocodewords of q-ary LDPC codes are examined and the weight of a pseudocodeword on the q-ary symmetric channel is defined.  ...  In [3] and [4] , the performance of non-binary LDPC codes, defined over larger finite fields and over integer rings, is investigated and compared with that of binary LDPC codes.  ... 
doi:10.1109/isit.2006.262072 dblp:conf/isit/KelleySR06 fatcat:o2wmymxos5ai7g6ukpbhh4ic4y

On the pseudocodeword weight and parity-check matrix redundancy of linear codes

Christine A. Kelley, Deepak Sridhara
2007 2007 IEEE Information Theory Workshop  
We provide some upper bounds on the BSCpseudoweight redundancy and illustrate the concept with some results for Hamming codes, tree-based and finite geometry LDPC codes, Reed-Muller codes and Hadamard  ...  Moreover, unlike the minimum distance which is unique to the code regardless of representation, the set of pseudocodewords, and therefore also the minimum pseudocodeword weight, depends on the graph representation  ...  We point out that this does not contradict the above theorem, as non-codeword pseudocodewords may also assume weights ≥ d(C). Theorem 2.1 says that any binary linear code has φ(C) ≤ 2 n−k .  ... 
doi:10.1109/itw.2007.4313040 fatcat:xlduof3m7rg6nnhv5bqjfdjjfm

Pseudocodewords of Tanner graphs [article]

Christine A. Kelley, Deepak Sridhara
2007 arXiv   pre-print
Lower bounds on the minimum pseudocodeword weight are presented for the BEC, BSC, and AWGN channel.  ...  weight.  ...  Let G represent a binary LDPC code C with minimum distance d min . Then G is called a Tanner graph (or, LDPC constraint graph) of C.  ... 
arXiv:cs/0504013v4 fatcat:gfxhs5yl75akpjqp3ie7s5o5d4

Pseudocodewords of Tanner Graphs

Christine A. Kelley, Deepak Sridhara
2007 IEEE Transactions on Information Theory  
Lower bounds on the minimum pseudocodeword weight are presented for the BEC, BSC, and AWGN channel.  ...  weight.  ...  Example 3.1: Some graphs with known t-values are: t ≤ 2 for cycle codes [1] , [2] , t = 1 for LDPC codes whose Tanner graphs are trees, and t ≤ 2 for LDPC graphs having a single cycle, and t ≤ s for  ... 
doi:10.1109/tit.2007.907501 fatcat:y76otn7b45hw3a22v3ckfluxzm

Tree-Based Construction of LDPC Codes Having Good Pseudocodeword Weights [article]

Christine Kelley, Deepak Sridhara, Joachim Rosenthal
2006 arXiv   pre-print
We also present some bounds on pseudocodeword weight for p-ary LDPC codes.  ...  Treating these codes as p-ary LDPC codes rather than binary LDPC codes improves their rates, minimum distances, and pseudocodeword weights, thereby giving a new importance to the finite geometry LDPC codes  ...  Moreover, we expect that p-ary LDPC codes to have larger minimum pseudocodeword weights than corresponding binary LDPC codes.  ... 
arXiv:cs/0510009v2 fatcat:jwxiqhje6ncqjitrsurvg3a73y

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.  ...  Furthermore, Tanner's parity-oriented eigenvalue lower bound on the minimum distance is generalized to yield a new lower bound on the minimum pseudocodeword weight.  ...  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

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.  ...  Furthermore, Tanner's parity-oriented eigenvalue lower bound on the minimum distance is generalized to yield a new lower bound on the minimum pseudocodeword weight.  ...  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

LP Decoding [chapter]

Jonathan Feldman
2008 Encyclopedia of Algorithms  
We also discuss the application of LP decoding to binary linear codes. We define the notion of a relaxation being symmetric for a binary linear code.  ...  We describe the notion of pseudocodewords under LP decoding, unifying many known characterizations for specific codes and channels.  ...  The fractional distance of a -symmetric polytope È for a binary linear code is equal to the minimum weight of a non-zero vertex of È.  ... 
doi:10.1007/978-0-387-30162-4_216 fatcat:4hhkbkxsfjahvi255da774qm7a

LP Decoding [chapter]

Jonathan Feldman
2016 Encyclopedia of Algorithms  
We also discuss the application of LP decoding to binary linear codes. We define the notion of a relaxation being symmetric for a binary linear code.  ...  We describe the notion of pseudocodewords under LP decoding, unifying many known characterizations for specific codes and channels.  ...  The fractional distance of a C-symmetric polytope P for a binary linear code C is equal to the minimum weight of a non-zero vertex of P.  ... 
doi:10.1007/978-1-4939-2864-4_216 fatcat:bce4u46mbrggjg2hdqkslrtqke

Non-Binary Protograph-Based LDPC Codes: Enumerators, Analysis, and Designs

Lara Dolecek, Dariush Divsalar, Yizeng Sun, Behzad Amiri
2014 IEEE Transactions on Information Theory  
We consider both random and constrained edgeweight labeling, and refer to the former as the unconstrained non-binary protograph-based LDPC codes (U-NBPB codes) and the latter as the constrained non-binary  ...  This paper provides a comprehensive analysis of non-binary low-density parity check (LDPC) codes built out of protographs.  ...  Ben-Yue Chang for help with the original formulation of NB-EXIT tool and for help with the design methodology of finite-length NBPB codes.  ... 
doi:10.1109/tit.2014.2316215 fatcat:72qisp7jn5abrghxtg2fkaihwy

Average min-sum decoding of LDPC codes

Nathan Axvig, Deanna Dreher, Katherine Morrison, Eric Psota, Lance C. Perez, Judy L. Walker
2008 2008 5th International Symposium on Turbo Codes and Related Topics  
Finally, AMS pseudocodewords are introduced and analyzed and their relationship to graph cover and LP pseudocodewords is explored, with particular focus on the AMS pseudocodewords of regular LDPC codes  ...  and cycle codes.  ...  In particular, for all non-codeword outputs in the range shown, the binary value assigned to the fourth coordinate is 1.  ... 
doi:10.1109/turbocoding.2008.4658725 fatcat:mecdixehcff75ldp4lr5uz2l4a

Using Linear Programming to Decode Binary Linear Codes

J. Feldman, M.J. Wainwright, D.R. Karger
2005 IEEE Transactions on Information Theory  
We describe a method for approximate ML decoding of an arbitrary binary linear code, based on a linear programming (LP) relaxation that is defined by a factor graph or parity check representation of the  ...  code.  ...  Fractional Distance A classical quantity associated with a code is its distance, which for a linear code is equal to the minimum weight of any non-zero codeword.  ... 
doi:10.1109/tit.2004.842696 fatcat:qf3estqpnvcqdocjtlnjw7d3cm

Guessing Facets: Polytope Structure and Improved LP Decoding [article]

Alexandros G. Dimakis, Amin A. Gohari, Martin J. Wainwright
2007 arXiv   pre-print
We begin by showing that for expander codes, every fractional pseudocodeword always has at least a constant fraction of non-integral bits.  ...  We then prove that for expander codes, the active set of any fractional pseudocodeword is smaller by a constant fraction than the active set of any codeword.  ...  Previous work by Koetter and Vontobel [9] established that for any bit-check regular LDPC code, there exist pseudocodewords for the additive white Gaussian noise (AWGN) channel with sublinear weight.  ... 
arXiv:0709.3915v1 fatcat:yl2tmsv67bcgvby6fzpzjp6ygm

Minimum Pseudo-Weight and Minimum Pseudo-Codewords of LDPC Codes [article]

Shu-Tao Xia, Fang-Wei Fu
2006 arXiv   pre-print
Then, we show that the lower bound of Kashyap and Vardy on the stopping distance of an LDPC code is also a lower bound on the pseudo-weight of a pseudo-codeword of this LDPC code with girth 4, and this  ...  Using these results we further show that for some LDPC codes, there are no other minimum pseudo-codewords except the real multiples of minimum codewords.  ...  Sridhara for kindly affording the preprint [10] , and Mr. X. Ge for his calculation of the minimum distance in Example 6 by computer.  ... 
arXiv:cs/0606051v2 fatcat:rremtpnt7vab3liovte22kpt34

Free Pseudodistance Growth Rates for Spatially Coupled LDPC Codes over the BEC [article]

Cunlu Zhou, David G. M. Mitchell, Roxana Smarandache
2018 arXiv   pre-print
In this paper, we consider ensembles of periodically time-varying spatially coupled LDPC (SC-LDPC) codes and the pseudocodewords arising from their finite graph covers of a fixed degree.  ...  We show that for certain (J,K)-regular SC-LDPC code ensembles and a fixed cover degree, the typical minimum pseudoweight of the unterminated (and associated tail-biting/terminated) SC-LDPC code ensembles  ...  The pseudodistance, w m min , for finite covers of a fixed degree m of G is defined as the minimum pseudoweight among all non-zero pseudocodewords from the degree-m covers.  ... 
arXiv:1809.04253v1 fatcat:lexailglpvgivirkb74e7jpmra
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