Explicit Construction of Families of LDPC Codes With No&amp;lt;tex&amp;gt;$4$&amp;lt;/tex&amp;gt;-Cycles

Kim, Jon-Lark; Peled, Uri N.; Perepelitsa, I. G.; Pless, Vera; Friedland, Shmuel

doi:10.1109/tit.2004.834760

Cited by 79 publications

(70 citation statements)

References 10 publications

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“…These graphs have yielded high performance LDPC codes. For example, recently, using D(m, q) as the Tanner graph, Kim et al [19] constructed and studied the associated LDPC code LU (m, q). Sin and Xiang [29] determined the dimension of the code LU (3, q) for odd q, and Anslan [2] very recently settled the case of even q.…”

Section: Examples Of Bipartite Graphs Attaining the Bound On μmentioning

confidence: 99%

Eigenvalues and expansion of bipartite graphs

Høholdt

Janwa

2012

Des. Codes Cryptogr.

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We prove lower bounds on the largest and second largest eigenvalue of the adjacency matrix of connected bipartite graphs and give necessary and sufficient conditions for equality. We give several examples of classes of graphs that are optimal with respect to the bounds. We prove that BIBD-graphs are characterized by their eigenvalues. Finally we present a new bound on the expansion coefficient of (c, d)-regular bipartite graphs and compare that with with a classical bound.

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Section: Examples Of Bipartite Graphs Attaining the Bound On μmentioning

confidence: 99%

Eigenvalues and expansion of bipartite graphs

Høholdt

Janwa

2012

Des. Codes Cryptogr.

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“…As an example, consider the [7,4,3] Hamming code, C [7,4,3] , with d ⊥ = 4. Any parity check matrix for C [7,4,3] contains the 7 nonzero binary vectors of length 3 and is thus isomorphic (under permutation of columns) to [12]: The Tanner graph corresponding to H [7,4,3] clearly contains 3 4-cycles. Indeed, for this code:…”

Section: (2)mentioning

confidence: 99%

“…precluding the existence of a 4-cycle-free Tanner graph while requiring that any Tanner graph for H [7,4,3] contains at least 2 4-cycles.…”

Section: (2)mentioning

confidence: 99%

“…It is now widely accepted that there is a relationship between the graph theoretic properties of a graphical code model and the performance of the iterative decoding algorithm implied by that model. Specifically, a number of authors have stressed the importance of designing LDPC codes with Tanner graphs that have no cycles of length four [2], [3], [4]. Furthermore, a number of authors have noted that Tanner graphs for classical linear block codes tend to contain 4-cycles and have thus investigated techniques for obtaining 4-cycle-free graphical models based on generalized parity check matrices [5], [6].…”

Section: Introductionmentioning

confidence: 99%

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Which Codes Have$4$-Cycle-Free Tanner Graphs?

Halford

Grant²,

Chugg³

2006

IEEE Trans. Inform. Theory

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“…One is based on permutation matrices [3], [27]. Others are based on Ramanujan graphs [19], [23], expander graphs [25], and q-regular bipartite graphs [12].…”

Section: Introductionmentioning

confidence: 99%

Small weight codewords in the LDPC codes arising from linear representations of geometries

Pepe

Storme

Voorde

2008

J of Combinatorial Designs

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In this paper, we investigate the minimum distance and small weight codewords of the LDPC codes of linear representations, using only geometrical methods. First we present a new lower bound on the minimum distance and we present a number of cases in which this lower bound is sharp. Then we take a closer look at the cases T * 2 (Θ) and T * 2 (Θ) D with Θ a hyperoval, hence q even, and characterize codewords of small weight. When investigating the small weight codewords of T * 2 (Θ) D , we deal with the case of Θ a regular hyperoval, i.e. a conic and its nucleus, separately, since in this case, we have a larger upper bound on the weight for which the results are valid.

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Explicit Construction of Families of LDPC Codes With No<tex>$4$</tex>-Cycles

Cited by 79 publications

References 10 publications

Eigenvalues and expansion of bipartite graphs

Eigenvalues and expansion of bipartite graphs

Which Codes Have$4$-Cycle-Free Tanner Graphs?

Small weight codewords in the LDPC codes arising from linear representations of geometries

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