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CiiT international journal of fuzzy systems

Construction of Optimal Puncturing of Low-Density Parity-Check Codes

S. Suresh Kumar, Manju Rajaram

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

Low-density parity-check with Length-compatible and low error floors (LDPC-LCLEF) on the binary symmetric channel is presented in this paper. These codes can support a wide range of lengths for a given rate and also the Tanner graphs are free of small trapping sets. This can be obtained by applying the schemes of shortening and puncturing to good LDPC codes and the trapping sets are selected using the trapping set ontology based on their relative harmfulness for the given decoding algorithm. The conventional approach does not guarantee to have a good performance as the degree distributions may be wildly varied. In this paper, a modified algorithm is presented to generate LDPC codes of a given rate with almost their degree distributions are same as the mother code degree distribution. Also the relative harmfulness for different trapping sets are evaluated and decoding failures of one trapping set in presence or absence of other trapping sets are analyzed. To facilitate this discussion, a new description of structured LDPC codes with arrays of permutation matrices as parity-check matrices is given. Numerical results show that the LDPC code constructed by this approach performed much better than the conventional approach.

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Low-density parity-check with Length-compatible and low error floors (LDPC-LCLEF) on the binary symmetric channel is presented in this paper. These codes can support a wide range of lengths for a given rate and also the Tanner graphs are free of small trapping sets. This can be obtained by applying the schemes of shortening and puncturing to good LDPC codes and the trapping sets are selected using the trapping set ontology based on their relative harmfulness for the given decoding algorithm. The conventional approach does not guarantee to have a good performance as the degree distributions may be wildly varied. In this paper, a modified algorithm is presented to generate LDPC codes of a given rate with almost their degree distributions are same as the mother code degree distribution. Also the relative harmfulness for different trapping sets are evaluated and decoding failures of one trapping set in presence or absence of other trapping sets are analyzed. To facilitate this discussion, a new description of structured LDPC codes with arrays of permutation matrices as parity-check matrices is given. Numerical results show that the LDPC code constructed by this approach performed much better than the conventional approach.
We present a method to construct low-density parity-check (LDPC) codes with low error floors on the binary symmetric channel. Codes are constructed so that their Tanner graphs are free of certain small trapping sets. These trapping sets are selected from the trapping set ontology for the Gallager A/B decoder. They are selected based on their relative harmfulness for a given decoding algorithm. We evaluate the relative harmfulness of different trapping sets for the sum–product algorithm by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets. We apply this method to construct structured LDPC codes. To facilitate the discussion, we give a new description of structured LDPC codes whose parity-check matrices are arrays of permutation matrices. This description uses Latin squares to define a set of permutation matrices that have disjoint support and to derive a simple necessary and sufficient condition for the Tanner graph of a code to be free of four cycles.
This paper is concerned with the construction of low-density parity-check (LDPC) codes with low error floors. Two main contributions are made. First, a new class of structured LDPC codes is introduced. The parity check matrices of these codes are arrays of permutation matrices which are obtained from Latin squares and form a finite field under some matrix operations. Second, a method to construct LDPC codes with low error floors on the binary symmetric channel (BSC) is presented. Codes are constructed so that their Tanner graphs are free of certain small trapping sets. These trapping sets are selected from the Trapping Set Ontology for the Gallager A/B decoder. They are selected based on their relative harmfulness for a given decoding algorithm. We evaluate the relative harmfulness of different trapping sets for the sum product algorithm (SPA) by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets.
Error correcting codes are used in virtually all communication systems to ensure reliable transmission of information. In 1948, Shannon established an upper-bound on the maximum rate at which information can be transmitted reliably over a noisy channel. Reliably transmitting information with a rate close to this theoretical limit, known as the channel capacity, has been the goal of channel coding scientists for the last five decades. The rediscovery of low-density parity-check (LDPC) codes in the 1990s added much-renewed excitement in the coding community. LDPC codes are interesting because they can approach channel capacity under sub-optimum decoding algorithms whose complexity is linear in the code length. Unsurprisingly, LDPC codes quickly attained their popularity in practical applications such as magnetic storage, wireless and optical communications. One, if not the most, important and challenging problem in LDPC code research is the study and analysis of the error floor phenomenon. This phenomenon is described as an abrupt degradation in the frame error rate performance of LDPC codes in the high signal-to-noise ratio region. Error floor is harmful because its presence prevents the LDPC decoder from reaching very low probability of decoding failure, an important requirement for many applications. Not long after the rediscovery of LDPC codes, scientists established that error floor is caused by certain harmful structures, most commonly known as trapping sets, in the Tanner representation of a code. Since then, the study of error floor mostly consists of three major problems: 1) estimating error floor; 2) constructing LDPC codes with low error floor and 3) designing decoders that are less susceptible to error floor. Although some parts of this dissertation can be used as important elements in error floor estimation, our main contributions are a novel method for constructing LDPC codes with low error floor and a novel class of low complexity decoding algorithms that can collectively alleviate error floor. These contributions are summarized as follows. A method to construct LDPC codes with low error floors on the binary symmetric channel is presented. Codes are constructed so that their Tanner graphs are free of certain small trapping sets. These trapping sets are selected from the Trapping Set Ontology for the Gallager A/B decoder. They are selected based on their relative harmfulness for a given decoding algorithm. We evaluate the relative harmfulness of different trapping sets for the sum-product algorithm by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets. We apply this method to construct structured LDPC codes. To facilitate the discussion, we give a new description of structured LDPC codes whose parity-check matrices are arrays of permutation matrices. This description uses Latin squares to define a set of permutation matrices that have disjoint support and to derive a simple necessary and sufficient condition for the Tanner graph of a code to be free of four-cycles. A new class of bit flipping algorithms for LDPC codes over the binary symmetric channel is proposed. Compared to the regular (parallel or serial) bit flipping algorithms, the proposed algorithms employ one additional bit at a variable node to represent its "strength." The introduction of this additional bit allows an increase in the guaranteed error correction capability. An additional bit is also employed at a check node to capture information which is beneficial to decoding. A framework for failure analysis and selection of two-bit bit flipping algorithms is provided. The main component of this framework is the (re)definition of trapping sets, which are the most "compact" Tanner graphs that cause decoding failures of an algorithm. A recursive procedure to enumerate trapping sets is described. This procedure is the basis for selecting a collection of algorithms that work well together. It is demonstrated that decoders which employ a properly selected group of the proposed algorithms operating in parallel can offer high speed and low error floor decoding.
In this paper, we study rate-compatible puncturing of finite-length Low-Density Parity-Check (LDPC) codes. First, we derive simple and yet good bounds on the expected performance of punctured codes (constructed by random puncturing) over Binary Erasure Channel (BEC) as a function of the performance of their parent LDPC code. We then present a novel rate-compatible puncturing scheme that is very easy to implement. Our scheme uses the idea that a more uniform distribution of punctured bits across the Tanner graph results in punctured codes with better performance. Although the puncturing scheme tailored to regular codes is presented, it is also directly applicable to irregular parent ensembles. By simulations, the proposed rate-compatible puncturing scheme is shown to be superior to the existing puncturing methods for both regular and irregular LDPC codes over BEC and Additive White Gaussian Noise (AWGN) Channel.
This paper introduces a class of structured low-density parity-check (LDPC) codes whose parity check matrices are arrays of permutation matrices. The permutation matrices are obtained from Latin squares and form a finite field under some matrix operations. They are chosen so that the Tanner graphs do not contain subgraphs harmful to iterative decoding algorithms. The construction of column-weight-three codes is presented. Although the codes are optimized for the Gallager A/B algorithm over the binary symmetric channel (BSC), their error performance is very good on the additive white Gaussian noise channel (AWGNC) as well.
We introduce a method to construct regular column-weight-three low-density parity-check (LDPC) codes with low error floors for the sum product algorithm (SPA) on the binary symmetric channel (BSC). The Tanner graphs of these codes are free of certain small trapping sets. These trapping sets are selected from the Trapping Set Ontology for the Gallager A/B decoder and are selected based on their relative harmfulness for the SPA. We evaluate the relative harmfulness of different trapping sets for the SPA by using the topological relations among them and by analyzing decoding failures on one trapping set in the presence or absence of other trapping sets. To the best of our knowledge, these codes outperform the best known short length, regular column-weight-three LDPC codes.