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Linear Time Encoding of LDPC Codes

Lu Jin, José M. F. Moura

Published Feb 27, 2010• 29 citations
OpenAlex

Abstract

Abstract—In this paper, we propose a linear complexity encoding method for arbitrary LDPC codes. We start from a simple graph-based encoding method “label-and-decide. ” We prove that the “label-and-decide ” method is applicable to Tanner graphs with a hierarchical structure—pseudo-trees—and that the resulting encoding complexity is linear with the code block length. Next, we define a second type of Tanner graphs—the encoding stopping set. The encoding stopping set is encoded in linear complexity by a revised label-and-decide algorithm—the “label-decide-recompute.” Finally, we prove that any Tanner graph can be partitioned into encoding stopping sets and pseudo-trees. By encoding each encoding stopping set or pseudo-tree sequentially, we develop a linear complexity encoding method for general low-density parity-check (LDPC) codes where the encoding complexity is proved to be less than 4 1 M 1 (k 0 1), where M is the number of independent rows in the parity-check matrix and k represents the mean row weight of the parity-check matrix. Index Terms—Encoding stopping set, low-density parity-check (LDPC) codes, linear complexity encoding, pseudo-tree, Tanner graphs.

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Low-density parity-check (LDPC) codes are a class of linear block codes with implementable decoders, which provide near-capacity performance on a large set of data-transmission and data-storage channels. LDPC codes were invented by Gallager in his 1960 doctoral dissertation and were mostly ignored during the 35 years that followed. One notable exception is the important work of Tanner in 1981, in which Tanner generalized LDPC codes and introduced a graphical representation of LDPC codes, now called a Tanner graph. The study of LDPC codes was resurrected in the mid 1990s with the work of MacKay, Luby, and others [3–6], who noticed, apparently independently of Gallager's work, the advantages of linear block codes with sparse (low-density) parity-check matrices. This chapter introduces LDPC codes and creates a foundation for further study of LDPC codes in later chapters. We start with the fundamental representations of LDPC codes via parity-check matrices and Tanner graphs. We then learn about the decoding advantages of linear codes that possess sparse parity-check matrices. We will see that this sparseness characteristic makes the code amenable to various iterative decoding algorithms, which in many instances provide near optimal performance. Gallager of course recognized the decoding advantages of such low-density parity-check codes and he proposed a decoding algorithm for the BI-AWGNC and a few others for the BSC. These algorithms have received much scrutiny in the past decade, and are still being studied.

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Cambridge University Press eBooks•Nov 19, 2009•5 citations
Abstract—In this paper, we propose a linear complexity encoding method for arbitrary LDPC codes. We start from a simple graph-based encoding method “label-and-decide. ” We prove that the “label-and-decide ” method is applicable to Tanner graphs with a hierarchical structure—pseudo-trees—and that the resulting encoding complexity is linear with the code block length. Next, we define a second type of Tanner graphs—the encoding stopping set. The encoding stopping set is encoded in linear complexity by a revised label-and-decide algorithm—the “label-decide-recompute.” Finally, we prove that any Tanner graph can be partitioned into encoding stopping sets and pseudo-trees. By encoding each encoding stopping set or pseudo-tree sequentially, we develop a linear complexity encoding method for general low-density parity-check (LDPC) codes where the encoding complexity is proved to be less than 4 1 M 1 (k 0 1), where M is the number of independent rows in the parity-check matrix and k represents the mean row weight of the parity-check matrix. Index Terms—Encoding stopping set, low-density parity-check (LDPC) codes, linear complexity encoding, pseudo-tree, Tanner graphs.
In this paper, we propose a linear complexity encoding method for arbitrary LDPC codes. We start from a simple graph-based encoding method ¿label-and-decide.¿ We prove that the ¿label-and-decide¿ method is applicable to Tanner graphs with a hierarchical structure-pseudo-trees-and that the resulting encoding complexity is linear with the code block length. Next, we define a second type of Tanner graphs-the encoding stopping set. The encoding stopping set is encoded in linear complexity by a revised label-and-decide algorithm-the ¿label-decide-recompute.¿ Finally, we prove that any Tanner graph can be partitioned into encoding stopping sets and pseudo-trees. By encoding each encoding stopping set or pseudo-tree sequentially, we develop a linear complexity encoding method for general low-density parity-check (LDPC) codes where the encoding complexity is proved to be less than4 ·M·((k¿- 1), whereMis the number of independent rows in the parity-check matrix andk¿represents the mean row weight of the parity-check matrix.
Low density parity check (LDPC) codes have stimulated a lot of interest in recent years due to their capacity approaching performance and availability of fast decoding algorithms. The existing encoding scheme for LDPC code usually incurs too high complexity and should be changed to encoding scheme with low complexity. An elementary turbo structured Tanner graph construction and encoding scheme for a family of regular LDPC codes attaining desired high girth is studied in this paper. The construction provides flexibility in choice of parameters like column weight (j > =2), row weight k and arbitrary girth g. Turbo structured low density parity check ( TS-LDPC) codes are regular LDPC codes with their Tanner graph consists of two trees connected by an interleaver. Linear complexity is attained with few alterations and restrictions of in Tanner graph of turbo structured LDPC codes. Simulation results demonstrates that not only the bit error rate (BER) performance of TSLDPC codes at low signal to noise ratio (SNR) is comparable with error performance of corresponding random LDPC codes, with better error performance at high SNR but linear encoding considerably reduces simulation time.
Low-density parity-check (LDPC) codes are a class of linear block codes with implementable decoders, which provide near-capacity performance on a large set of data-transmission and data-storage channels. LDPC codes were invented by Gallager in his 1960 doctoral dissertation and were mostly ignored during the 35 years that followed. One notable exception is the important work of Tanner in 1981, in which Tanner generalized LDPC codes and introduced a graphical representation of LDPC codes, now called a Tanner graph. The study of LDPC codes was resurrected in the mid 1990s with the work of MacKay, Luby, and others [3–6], who noticed, apparently independently of Gallager's work, the advantages of linear block codes with sparse (low-density) parity-check matrices. This chapter introduces LDPC codes and creates a foundation for further study of LDPC codes in later chapters. We start with the fundamental representations of LDPC codes via parity-check matrices and Tanner graphs. We then learn about the decoding advantages of linear codes that possess sparse parity-check matrices. We will see that this sparseness characteristic makes the code amenable to various iterative decoding algorithms, which in many instances provide near optimal performance. Gallager of course recognized the decoding advantages of such low-density parity-check codes and he proposed a decoding algorithm for the BI-AWGNC and a few others for the BSC. These algorithms have received much scrutiny in the past decade, and are still being studied.
An algorithm for efficient encoding of LDPC codes is presented that does not impose any restrictions on the construction of the parity-check matrices. The algorithm modifies the parity check matrix, without changing the subspace spanned by its rows, by removing linear dependent rows and adding a small number of new rows such that the graph-based message-passing encoder will not get stuck in a stopping set. The added rows are designed by a new algorithm which is based on the notion of the "key set". The encoder exploits the sparseness of the parity-check matrix, and the encoding complexity grows almost linear with the blocksize, because the number of added rows, which may not be sparse, is relatively small.
Introduction In this chapter we introduce low-density parity-check (LDPC) codes, a class of error correction codes proposed by Gallager in his 1962 PhD thesis 12 years after error correction codes were first introduced by Hamming (published in 1950). Both Hamming codes and LDPC codes are block codes: the messages are broken up into blocks to be encoded at the transmitter and similarly decoded as separate blocks at the receiver. While Hamming codes are short and very structured with a known, fixed, error correction ability, LDPC codes are the opposite, usually long and often constructed pseudo-randomly with only a probabilistic notion of their expected error correction performance. The chapter begins by presenting parity bits as a means to detect and, when more than one is employed, to correct errors in digital data. Block error correction codes are described as a linear combination of parity-check equations and thus defined by their parity-check matrix representation. The graphical representation of codes by Tanner graphs is presented and the necessary graph theoretic concepts introduced. In Section 2.4 iterative decoding algorithms are introduced using a hard decision algorithm (bit flipping), so that the topic is developed first without reference to probability theory. Subsequently the sum–product decoding algorithm is presented. This chapter serves as a self-contained introduction to LDPC codes and their decoding. It is intended that the material presented here will enable the reader to implement LDPC encoders and iterative decoders.