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Allerton Conference on Communication, Control, and Computing

Analysis of bipartite graph codes on the binary erasure channel

Arya Mazumdar

Published Jan 1, 2007• 1 citation
OpenAlex

Abstract

We derive density evolution equations for codes on bipartite graphs (BG codes) for the binary erasure channel (BEC).We study the cases of local codes being introduced on only one side of the graph (Generalized LDPC codes) as well as on both sides.Each local code is assumed to correct up to a number of erasures one less than its distance.We define and enumerate stopping sets for BG codes, which serves an important tool for analysis of decoding thresholds on the BEC.

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IEEE Transactions on Information Theory•Mar 15, 2011•10 citations
We derive density evolution equations for codes on bipartite graphs (BG codes) for the binary erasure channel (BEC).We study the cases of local codes being introduced on only one side of the graph (Generalized LDPC codes) as well as on both sides.Each local code is assumed to correct up to a number of erasures one less than its distance.We define and enumerate stopping sets for BG codes, which serves an important tool for analysis of decoding thresholds on the BEC.
In this paper we present a thorough analysis of non binary LDPC codes over the binary erasure channel. First, the decoding of non binary LDPC codes is investigated. The proposed algorithm performs ldquoon-the flyrdquo decoding, i.e. it starts decoding as soon as the first symbols are received, which generalizes the erasure decoding of binary LDPC codes. Next, we evaluate the asymptotical performance of ensembles of non binary LDPC codes, by using the density evolution method. Density evolution equations are derived by taking into consideration both the irregularity of the bipartite graph and the probability distribution of the graph edge labels. Finally, infinite-length performance of some ensembles of non binary LDPC codes for different edge label distributions are shown.
In this paper, we first introduce the extended binary representation of non-binary codes, which corresponds to a covering graph of the bipartite graph associated with the non-binary code. Then we show that non-binary codewords correspond to binary codewords of the extended representation that further satisfy some simplex-constraint: that is, bits lying over the same symbol-node of the non-binary graph must form a codeword of a simplex code. Applied to the binary erasure channel (BEC), this description leads to a binary erasure decoding algorithm of non-binary LDPC codes, whose complexity depends linearly on the cardinality of the alphabet. We also give insights into the structure of stopping sets for non-binary LDPC codes, and discuss several aspects related to upper-layer FEC applications.
In this correspondence, we first investigate some analytical aspects of the recently proposed improved decoding algorithm for low-density parity-check (LDPC) codes over the binary erasure channel (BEC). We derive a necessary and sufficient condition for the improved decoding algorithm to successfully complete decoding when the decoder is initialized to guess a predetermined number of guesses after the standard message-passing terminates at a stopping set. Furthermore, we present improved bounds on the number of bits to be guessed for successful completion of the decoding process when a stopping set is encountered. Under suitable conditions, we derive a lower bound on the number of iterations to be performed for complete decoding of the stopping set. We then present a superior, novel improved decoding algorithm for LDPC codes over the binary erasure channel (BEC). The proposed algorithm combines the observation that a considerable fraction of unsatisfied check nodes in the neighborhood of a stopping set are of degree two, and the concept of guessing bits to perform simple and intuitive graph-theoretic manipulations on the Tanner graph. The proposed decoding algorithm has a complexity similar to previous improved decoding algorithms. Finally, we present simulation results of short-length codes over BEC that demonstrate the superiority of our algorithm over previous improved decoding algorithms for a wide range of bit error rates
The authors show that a popular way of constructing quasi-cyclic low-density parity check (LDPC) codes is a special case of a construction which is common in graph theory and group theory. It is shown that a generalisation of this construction as coset graphs produces (dv, dc)-regular LDPC codes that have an advantage in terms of the minimum stopping set size compared to quasi-cyclic LDPC codes. A (dv, dc)-regular quasi-cyclic LDPC code cannot have minimum stopping set size larger than (dv+1)!. However, by using coset graphs, a (3, 5)-regular LDPC code with minimum stopping set size of 28 and a (3, 4)-regular LDPC code with minimum stopping set size larger than 32 have been obtained. In addition, the idea of coset graphs also provides a compact algebraic way of describing bipartite graph and the associated parity-check matrix of an LDPC code. Simulation results of iterative decoding of the coset graphs LDPC codes over the binary erasure channel show that some of the codes converge well and based on the truncated stopping set distributions of the codes, which are exhaustively and efficiently enumerated, the error-floor of the codes at low probability of erasure is estimated.
In this paper, the transmission over the binary erasure channel (BEC) using non-binary LDPC (NBLDPC) codes is considered. The concept of peeling decoder and stopping sets is generalized to NBLDPC codes. Using these generalizations, a combinatorial characterization of decoding failures of NBLDPC codes is given, under assumption that the Belief Propagation (BP) decoder is used. Then, the residual ensemble of codes resulted by the BP decoder is defined and the design rate and the expectation of total number of codewords of the residual ensemble are computed. The decoding failure criterion combined with the density evolution analysis helps us to compute the asymptotic residual degree distribution for NBLDPC codes. Our approach to compute the residual degree distribution on the check node side is not efficient as it is based on enumeration of all the possible connections on the check node side which satisfy the decoding failure criterion. So, the computation of the asymptotic check node side residual degree distribution and further part of our analysis is performed for NBLDPC codes over GF2mwithm= 2 . In order to show that asymptotically almost every code in such LDPC ensemble has a rate equal to the design rate, we generalize the argument of the Maxwell construction to NBLDPC codes, defined over GF22. It is also observed that, like in the binary setting, the Maxwell construction, relating the performance of MAP and BP decoding holds in this setting.