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A Generalized Algebraic Approach to Optimizing SC-LDPC Codes
[article]

2017
*
arXiv
*
pre-print

In this paper, we unify all existing SC-LDPC code construction methods under

arXiv:1710.03619v1
fatcat:5qgv6hjwkncyng6iszpzcyr2eu
*a*new generalized description of SC-LDPC codes based on algebraic lifts of graphs. ... That is, σ 1*a*,x = max(φ 1*a*,x , θ 1x , θ 3x ), σ 1*a*,y = max(φ 1*a*,y , θ 1y , θ 3y ), σ 2*a*,x = min(φ 2*a*,x , θ 2x , θ 4x ), σ 2*a*,y = min(φ 2*a*,y , θ 2y , θ 4y ). ... These points are denoted by σ 1*a*and σ 2*a*. ...##
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Algebraic Constructions of Graph-Based Nested Codes from Protographs
[article]

2010
*
arXiv
*
pre-print

Nested codes have been employed in

arXiv:1006.2977v1
fatcat:ukszspict5e7dphabivrqkdwqq
*a*large number of communication applications as*a*specific case of superposition codes, for example to implement binning schemes in the presence of noise, in joint network-channel ... For j < k, we form the following j × k matrix P with elements from Z * m that has as its (s, t)th element P s,t = b s*a*t as follows: P = 1*a**a*2 . . .*a*k−1 b ab*a*2 b . . .*a*k−1 b . . . . . . ... . , n, exactly c j kj-cycles. 2 Here we distinguish between*a*k-cycle in*a*graph which is*a*closed walk of length k, and*a*cycle of*a*permutation which is*a*closed set of numbers in the cycle representation ...##
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Eigenvalue bounds on the pseudocodeword weight of expander codes
[article]

2007
*
arXiv
*
pre-print

Furthermore, Tanner's parity-oriented eigenvalue lower bound on the minimum distance is generalized to yield

arXiv:0708.2462v1
fatcat:5r7thuq535cjxnzrjwznougqpe
*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. ... Acknowledgments We thank Joachim Rosenthal and the reviewers for*a*careful proof-reading of this paper and their valuable comments. We believe their feedback has greatly improved the paper. ...##
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Geometric WOM codes and coding strategies for multilevel flash memories
[article]

2012
*
arXiv
*
pre-print

Using ideas from Merkx ('84), we present

arXiv:1206.5762v1
fatcat:fmm4tzcwrnd3fn7wxyq4ytzkim
*a*construction of WOM codes based on finite Euclidean geometries over F_2. ...*A*brief discussion of the average-write performance of these strategies, as well as concatenation methods for WOM codes is also provided. ... The points are the nonzero (m+1)tuples (*a*0 ,*a*1 , . . . ,*a*m ) ∈ F m+1 2 , and*a*line through two distinct points*a*0 and*a*1 contains exactly the set of points {*a*0 ,*a*1 ,*a*0 +*a*1 }. ...##
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Pseudocodewords of Tanner Graphs

2007
*
IEEE Transactions on Information Theory
*

This papers presents

doi:10.1109/tit.2007.907501
fatcat:y76otn7b45hw3a22v3ckfluxzm
*a*detailed analysis of pseudocodewords of Tanner graphs. ... pseudocodewords have components upper bounded by*a*finite value t that is dependent on the graph structure. ... Figure 2 shows*a*base graph G and*a*degree four cover of G.*A*codewordĉ in*a*lift graphĜ of*a*Tanner graph G is defined analogously as in Definition 2.1. ...##
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Multidimensional Decoding Networks for Trapping Set Analysis
[chapter]

2017
*
Lecture Notes in Computer Science
*

We present

doi:10.1007/978-3-319-66278-7_2
fatcat:ofio5c4mxrg3rk47dolmvbpq7y
*a*novel multidimensional network model as*a*means to analyze decoder failure and characterize trapping sets of graphbased codes. ... We identify*a*special class of these decoding networks, which we call transitive networks, and show how they may be used to identify trapping sets and inducing sets. ... Under Gallager*A*decoding, representation*A*does not yield*a*transitive decoding network. ...##
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LDPC codes from voltage graphs

2008
*
2008 IEEE International Symposium on Information Theory
*

If D(

doi:10.1109/isit.2008.4595095
dblp:conf/isit/KelleyW08
fatcat:fvttiwkphraghnormlagnhhwmq
*a*1 ,*a*2 ; b) is*a*subgraph of*a*simple bipartite graph with*a*1 ,*a*2 ≥ 1 and b ≥ 0, then*a*1 ,*a*2 ≥ 4 and are even. ... Then X, by Theorem 4.5, X has either*a*subgraph T (*a*, b, c) with 2a + 2b + 2c ≤ n or*a*subgraph D(*a*1 ,*a*2 ; b) with 2(*a*1 +*a*2 ) + 4b ≤ n. ...##
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Pseudocodewords of Tanner graphs
[article]

2007
*
arXiv
*
pre-print

This papers presents

arXiv:cs/0504013v4
fatcat:gfxhs5yl75akpjqp3ie7s5o5d4
*a*detailed analysis of pseudocodewords of Tanner graphs. ... pseudocodewords have components upper bounded by*a*finite value t that is dependent on the graph structure. ...*A*. ...##
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Variations of the McEliece Cryptosystem
[article]

2017
*
arXiv
*
pre-print

The first one is based on

arXiv:1612.05085v2
fatcat:iotsnlu6qzdqjpatxcv2vmvrqe
*a*relaxation of the column permutation in the classical McEliece scrambling process. ... These codes are known for their excellent error-correction performance and allow for*a*relatively low key size in the cryptosystem. ... For*a*vector*a*= (*a*1 , . . . ,*a*m ) ∈ F m denote by C*a*∈ F m×m the circulant matrix with i-th row given by the cyclic shift (*a*i , . . . ,*a*m ,*a*1 We will use G ′ (D) for the public key. ...##
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On the independence and domination numbers of replacement product graphs

2016
*
Involve. A Journal of Mathematics
*

The replacement product is

doi:10.2140/involve.2016.9.181
fatcat:yoq4krxo25e3raizb5knk4glaa
*a*noncommutative graph operation that has been widely applied in many areas. One of its advantages over other graph products is its ability to produce sparse graphs. ... Acknowledgements The results in this paper were part of Cummings' undergraduate honors thesis, "On invariants of replacement product graphs", that was conducted under the supervision of*Kelley*. ... Invariants of graphs, including the independence and domination numbers of*a*graph, have also been widely studied. Many applications in computer science and 182 JAY CUMMINGS AND*CHRISTINE**A*. ...##
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Extremal absorbing sets in low-density parity-check codes

2021
*
Advances in Mathematics of Communications
*

In this paper, we determine the sizes of absorbing sets that can occur in general and leftregular LDPC code graphs, with emphasis on the range of b for

doi:10.3934/amc.2021003
fatcat:5bqyowmvgjalnbc3aprwm4fg4a
*a*given*a*for which an (*a*, b)-absorbing set may exist ... We identify certain cases of extremal absorbing sets that are elementary,*a*particularly harmful class of absorbing sets, and also introduce the notion of minimal absorbing sets which will help in designing ... Acknowledgments We thank the anonymous reviewers for their helpful comments that improved the quality of the paper, including*a*nicer proof of Theorem 3.2. ...##
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Zig-zag and Replacement Product Graphs and LDPC Codes
[article]

2007
*
arXiv
*
pre-print

The performance of codes defined from graphs depends on the expansion property of the underlying graph in

arXiv:cs/0611155v2
fatcat:3ex3qwbknnhltmvod65dbbtrai
*a*crucial way. ...*A*modification of the zig-zag product is also introduced, which can operate on two unbalanced biregular bipartite graphs. ... is (*a*, b)(c, d) = (*a*+ φ b (c), b + d), for*a*, c ∈*A*, b, d ∈ B. ...##
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Algebraic constructions of graph-based nested codes from protographs

2010
*
2010 IEEE International Symposium on Information Theory
*

Nested codes have been employed in

doi:10.1109/isit.2010.5513628
dblp:conf/isit/KelleyK10
fatcat:d4c6h7yibvd2xopihsnpn73n4a
*a*large number of communication applications as*a*specific case of superposition codes, for example to implement binning schemes in the presence of noise, in joint network-channel ... For j < k, we form the following j × k matrix P with elements from Z * m that has as its (s, t)th element P s,t = b s*a*t as follows: P = 1*a**a*2 . . .*a*k−1 b ab*a*2 b . . .*a*k−1 b . . . . . . ... . , n, exactly c j kj-cycles. 2 Here we distinguish between*a*k-cycle in*a*graph which is*a*closed walk of length k, and*a*cycle of*a*permutation which is*a*closed set of numbers in the cycle representation ...##
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Motives for physical dating violence among college students: A gendered analysis

2015
*
Psychology of Violence
*

Method: The sample included 221 college students who reported

doi:10.1037/a0036171
fatcat:mmr5274l5fcqpe3utpjpikm6m4
*a*history of physical DV perpetration and completed surveys for course credit. ... This study explores this gap in the literature with*a*specific focus on gender, coping, DV perpetration and victimization, and attitudes toward violence. ... ) and who participated in*a*larger mixed-methodological study assessing students' perceptions of abuse in dating relationships (Edwards, Dardis,*Kelley*, & Gidycz, 2013) . ...##
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On codes designed via algebraic lifts of graphs

2008
*
2008 46th Annual Allerton Conference on Communication, Control, and Computing
*

In particular, we present

doi:10.1109/allerton.2008.4797704
fatcat:gptjyj4savd3zpsiyzw4pvai5q
*a*construction of LDPC codes by giving an algebraic method of choosing the permutation voltages and illustrate the usefulness of the proposed technique via simulation results. ... An (*a*1 ,*a*2 ; b)-dumbbell graph is*a*connected graph consisting of two disjoint cycles*A*1 and*A*2 of lengths*a*1 ≥ 1 and*a*2 ≥ 1, that are connected by*a*path B of length b ≥ 0. ... In [5] , it was shown that if voltages were assigned from an abelian group G to*a*base graph G, where G contained either*a*(*a*, b, c)theta graph or*a*(*a*1 ,*a*2 ; b)-dumbbell graph as*a*subgraph, 2*A*...
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