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18
Shortened array codes of large girth,” in
 IEEE Transactions on Information Theory
, 2006
"... Abstract — One approach to designing structured lowdensity paritycheck (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the paritycheck matrix contain all the variables involved in short cycles. This approach is especially effective i ..."
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Abstract — One approach to designing structured lowdensity paritycheck (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the paritycheck matrix contain all the variables involved in short cycles. This approach is especially effective if the paritycheck matrix of a code is a matrix composed of blocks of circulant permutation matrices, as is the case for the class of codes known as array codes. We show how to shorten array codes by deleting certain columns of their paritycheck matrices so as to increase their girth. The shortening approach is based on the observation that for array codes, and in fact for a slightly more general class of LDPC codes, the cycles in the corresponding Tanner graph are governed by certain homogeneous linear equations with integer coefficients. Consequently, we can selectively eliminate cycles from an array code by only retaining those columns from the paritycheck matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles. We provide Ramseytheoretic estimates for the maximum number of columns that can be retained from the original paritycheck matrix with the property that the sequence of their indices avoid solutions to various types of cyclegoverning equations. This translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles. Simulation results show that for the codes considered, shortening them to increase the girth can lead to significant gains in signaltonoise ratio in the case of communication over an additive white Gaussian noise channel. Index Terms — Array codes, LDPC codes, shortening, cyclegoverning equations
Which codes have 4cyclefree Tanner graphs
 IEEE Trans. Information Theory
, 2006
"... Abstract — Let C be an [n, k, d] binary linear code with rate R = k/n and dual C ⊥. In this work, it is shown that C can be represented by a 4cyclefree Tanner graph only if: pd ⊥ ≤ $r np(p − 1) + n2 ..."
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Cited by 7 (0 self)
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Abstract — Let C be an [n, k, d] binary linear code with rate R = k/n and dual C ⊥. In this work, it is shown that C can be represented by a 4cyclefree Tanner graph only if: pd ⊥ ≤ $r np(p − 1) + n2
LDPC Codes Based on Latin Squares: Cycle Structure, Stopping, and Trapping Set Analysis
"... It is well known that certain combinatorial structures in the Tanner graph of a lowdensity paritycheck code exhibit a strong influence on its performance under iterative decoding. These structures include cycles, stopping/trapping sets and parameters such as the diameter of the code. In general, i ..."
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Cited by 6 (1 self)
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It is well known that certain combinatorial structures in the Tanner graph of a lowdensity paritycheck code exhibit a strong influence on its performance under iterative decoding. These structures include cycles, stopping/trapping sets and parameters such as the diameter of the code. In general, it is very hard to find a complete characterization of such configurations in an arbitrary code, and even harder to understand the intricate relationships that exist between these entities. It is therefore of interest to identify a simple setting in which all the described combinatorial structures can be enumerated and studied within a joint framework. One such setting is developed in this paper, for the purpose of analyzing the distribution of short cycles and the structure of stopping and trapping sets in Tanner graphs of LDPC codes based on idempotent and symmetric Latin squares. The paritycheck matrices of LDPC codes based on Latin squares have a special form that allows for connecting combinatorial parameters of the codes with the number of certain subrectangles in the Latin squares. Subrectangles of interest can be easily identified, and in certain instances, completely enumerated. The presented study can be extended in several different directions, one of which is concerned with modifying the code design process in order to eliminate or reduce the number of configurations bearing a negative influence on the performance of the code. Another application of the results includes determining to which extent a configuration governs the behavior of the bit error rate (BER) curve in the waterfall and errorfloor regions.
Minimum pseudoweight and minimum pseudocodewords of LDPC codes
 IEEE Transactions on Information Theory
, 2008
"... In this correspondence, we study the minimum pseudoweight and minimum pseudocodewords of lowdensity paritycheck (LDPC) codes under linear programming (LP) decoding. First, we show that the lower bound of Kelly, Sridhara, Xu and Rosenthal on the pseudoweight of a pseudocodeword of an LDPC code ..."
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Cited by 4 (0 self)
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In this correspondence, we study the minimum pseudoweight and minimum pseudocodewords of lowdensity paritycheck (LDPC) codes under linear programming (LP) decoding. First, we show that the lower bound of Kelly, Sridhara, Xu and Rosenthal on the pseudoweight of a pseudocodeword of an LDPC code with girth greater than 4 is tight if and only if this pseudocodeword is a real multiple of a codeword. 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 pseudoweight of a pseudocodeword of this LDPC code with girth 4, and this lower bound is tight if and only if this pseudocodeword is a real multiple of a codeword. Using these results we further show that for some LDPC codes, there are no other minimum pseudocodewords except the real multiples of minimum codewords. This means that the LP decoding for these LDPC codes is asymptotically optimal in the sense that the ratio of the probabilities of decoding errors of LP decoding and maximumlikelihood decoding approaches to 1 as the signaltonoise ratio leads to infinity. Finally, some LDPC codes are listed to illustrate these results. Index Terms: LDPC codes, linear programming (LP) decoding, fundamental cone, pseudocodewords, pseudoweight, stopping sets.
LDPC codes from trianglefree line sets
 Designs, Codes, and Cryptog
, 2004
"... We study sets of lines of AG(n, q) and P G(n, q) with the property that no three lines form a triangle. As a result the associated pointline incidence graph contains no 6cycles and necessarily has girth at least 8. One can then use the associated incidence matrices to form binary linear codes whic ..."
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Cited by 4 (1 self)
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We study sets of lines of AG(n, q) and P G(n, q) with the property that no three lines form a triangle. As a result the associated pointline incidence graph contains no 6cycles and necessarily has girth at least 8. One can then use the associated incidence matrices to form binary linear codes which can be considered as LDPC codes. The relatively high girth allows for efficient implementation of these codes. We give two general constructions for such trianglefree line sets and give the parameters for the associated codes when q is small. 1
On a class of quasicyclic LDPC codes
, 2005
"... We study a class of quasicyclic LDPC codes. We provide both a Grobner basis approach, which leads to precise conditions on the code dimension, and a graph theoretic prospective, that lets us guarantee high girth in their Tanner graph. Experimentally, the codes we propose perform no worse than rando ..."
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Cited by 2 (2 self)
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We study a class of quasicyclic LDPC codes. We provide both a Grobner basis approach, which leads to precise conditions on the code dimension, and a graph theoretic prospective, that lets us guarantee high girth in their Tanner graph. Experimentally, the codes we propose perform no worse than random LDPC codes with their same parameters, which is a significant achievement for algebraic codes.
TSLDPC Codes: TurboStructured Codes With Large Girth
"... Abstract—We consider turbostructured lowdensity paritycheck (TSLDPC) codes—structured regular codes whose Tanner graph is composed of two trees connected by an interleaver. TSLDPC codes with good girth properties are easy to construct: careful design of the interleaver component prevents short c ..."
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Abstract—We consider turbostructured lowdensity paritycheck (TSLDPC) codes—structured regular codes whose Tanner graph is composed of two trees connected by an interleaver. TSLDPC codes with good girth properties are easy to construct: careful design of the interleaver component prevents short cycles of any desired length in its Tanner graph. We present algorithms to construct TSLDPC codes with arbitrary column weight � ! P and row weight � and arbitrary girth �. We develop a linear complexity encoding algorithm for a type of TSLDPC codes—encoding friendly TSLDPC (EFTSLDPC) codes. Simulation results demonstrate that the biterror rate (BER) performance at low signaltonoise ratio (SNR) is competitive with the error performance of random LDPC codes of the same size, with better error floor properties at high SNR. Index Terms—Error floor, girth, interleaver, lowdensity paritycheck (LDPC) codes, turbostructured.
Determination of the shortest balanced cycles in QCLDPC codes Matrix
"... Abstract—In this paper, we determinate the shortest balanced cycles of quasicyclic lowdensity paritycheck (QCLDPC) codes. We show the structure of balanced cycles and their necessary and sufficient existence conditions. Furthermore, we determine the shortest matrices of balanced cycle. Finally a ..."
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Cited by 1 (1 self)
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Abstract—In this paper, we determinate the shortest balanced cycles of quasicyclic lowdensity paritycheck (QCLDPC) codes. We show the structure of balanced cycles and their necessary and sufficient existence conditions. Furthermore, we determine the shortest matrices of balanced cycle. Finally all nonequivalent minimal matrices of the shortest balanced cycles are presented in this paper. Index Terms—Girth, quasicyclic lowdensity paritycheck (QCLDPC) codes, balanced cycles. I.
On the Gröbner basis of a family of quasicyclic LDPC codes
, 2005
"... In [6] a class of quasicyclic LDPC codes has been proposed, whose information rate is 1/2. We generalize that construction to arbitrary rates 1/s and we provide a Grobner basis for their dual codes, under a nonrestrictive condition. As a consequence, we are able to determine their dimension. 1 ..."
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In [6] a class of quasicyclic LDPC codes has been proposed, whose information rate is 1/2. We generalize that construction to arbitrary rates 1/s and we provide a Grobner basis for their dual codes, under a nonrestrictive condition. As a consequence, we are able to determine their dimension. 1
On codes generated from quadratic surfaces in PG(3, q)
, 2004
"... We construct two families of lowdensity paritycheck codes using pointline incidence structures in PG(3, q). The selection of lines for each structure relies on the geometry of the two classical quadratic surfaces in PG(3, q), the hyperbolic quadric and the elliptic quadric. ..."
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We construct two families of lowdensity paritycheck codes using pointline incidence structures in PG(3, q). The selection of lines for each structure relies on the geometry of the two classical quadratic surfaces in PG(3, q), the hyperbolic quadric and the elliptic quadric.