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International Conference on Communications

The Minimal Product Parity Check Matrix and Its Application

Morteza Esmaeili

Published Jun 1, 2006• 4 citations
Publisher

Abstract

In decoding a linear block code C by iterative decoding algorithms, these algorithms are essentially applied on a graphical representation of C such as Tanner graph and factor graph. Due to the impact of the structure of these graphs on the performance of code, we consider minimal parity check matrices(matrices with minimum number of nonzero entries) of product codes. An algorithm constructing such matrices is given. It turns out that under the given construction method, product coding technique is indeed a powerful method to construct irregular LDPC codes. Given two codes A and B, the construction method produces a Tanner graph with girth g := min {8, ga, gb} for the product code A ⊗ B, where ga and gb are the girth of Tanner graphs representing A and B, respectively. Simulation results confirm the positive practical impact of the minimality of the parity check matrices.

DOI
10.1109/icc.2006.254896

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In decoding a linear block code C by iterative decoding algorithms, these algorithms are essentially applied on a graphical representation of C such as Tanner graph and factor graph. Due to the impact of the structure of these graphs on the performance of code, we consider minimal parity check matrices(matrices with minimum number of nonzero entries) of product codes. An algorithm constructing such matrices is given. It turns out that under the given construction method, product coding technique is indeed a powerful method to construct irregular LDPC codes. Given two codes A and B, the construction method produces a Tanner graph with girth g := min {8, ga, gb} for the product code A ⊗ B, where ga and gb are the girth of Tanner graphs representing A and B, respectively. Simulation results confirm the positive practical impact of the minimality of the parity check matrices.
Low Density Parity Check (LDPC) codes are specified by the parity check matrix H. Smaller girths in the Tanner graphs of LDPC codes prevent the sum-product algorithm from converging, i.e., short cycles in H-matrix degrade the performance of LDPC decoder. In this paper, we present an algorithm for constructing LDPC codes without short cycles. The H-matrix must be sufficiently sparse to increase the girth of a code. In this work, we present a simple method to increase the sparsity of matrix and provide lemmas to avoid short cycles of lengths 4 and 6. We design sub-matrices for column-weight two and column-weight three codes. This design makes use of a simple shifting function given by the proposed scheme. These are then combined into a final H-matrix using Gray code representations. The proposed method provides structured, regular codes with flexibility in code rates and code lengths. The regularity of these codes gives the advantages like simplicity of hardware implementation and fast encoding. Key words LDPC codes, Tanner Graph, Girth, Gray codes
This paper gives a new random construction for LDPC codes, which uses an iterative algorithm to avoid short cycles in the Tanner graph. The construction method has low complexity and great flexible choice in LDPC code parameters including code length, code rate, least girth of the graph, and the weight of column and row in the parity-check matrix. The method can be applied to the irregular LDPC codes and strict regular LDPC codes. Simulations show that the method performs well with iterative decoding.
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.
Introduction The construction of binary low-density parity-check (LDPC) codes simply involves replacing a small number of the values in an all-zeros matrix by 1s in such a way that the rows and columns have the required degree distribution. In many cases, randomly allocating the entries in H will produce a reasonable LDPC code. However, the construction of H can affect the performance of the sum–product decoder, significantly so for some codes, and also the implementation complexity of the code. While there is no one recipe for a “good” LDPC code, there are a number of principles that inform the code designer. The first obvious decisions are which degree distribution to choose and how to construct the matrix with the chosen degrees, i.e. pseudo-randomly or with some sort of structure. Whichever construction is chosen, the features to consider include the girth of the Tanner graph and the minimum distance of the code. In this chapter we will discuss those properties of an LDPC code that affect its iterative decoding performance and then present the common construction methods used to produce codes with the preferred properties. Following common practice in the field we will call the selection of the degree distributions for an LDPC code code design and the methods to assign the locations in the parity-check matrix for the 1 entries code construction .
Only since the mid-1990s, with the increase in computational power, has the\nimportance been realized regarding low-density parity-check (LDPC) codes as leading error\ncorrecting codes. The work in this thesis primarily studies LDPC codes that are de???ned by a\nparity-check matrix comprised of r _ r permutation matrix blocks, that is, permutation-block\nLDPC codes. More specifically, we study their minimum Hamming distance, as this is a\nparameter that affects the performance of linear codes, and thus LDPC codes.\nIt has been proven that there are upper bounds on the minimum Hamming distance of\nquasi-cyclic codes, de???ned by a parity-check matrix comprised of r _ r circulant matrix\nblocks, regardless of how large we allow r. However, we show that if general permutation\nmatrices are used instead of circulant matrices, we are able to exceed the bounds allowed by\nquasi-cyclic codes. Graphical representations of the codes, called Tanner-graphs, are used in\nproving many useful results regarding permutation-block LDPC codes and their minimum\nHamming distance. We show a characterization of the minimum Hamming distance of these\ncodes and we produce an algorithm for constructing permutation-block LDPC codes with\nlarge minimum Hamming distance.
This paper presents a unifying framework to construct low-density parity-check (LDPC) codes with associated Tanner graphs of desired girth. Towards this goal, we highlight the role that a certain square matrix that appears in the product of the parity-check matrix with its transpose has in the construction of codes with graphs of desired girth and further explore it in order to generate the set of necessary and sufficient conditions for a Tanner graph to have a given girth between 6 and 12. For each such girth, we present algorithms to construct codes of the desired girth and we show how to use them to compute the minimum necessary value of the lifting factor. For girth larger than 12, we show how to use multi-step graph lifting methods to deterministically modify codes in order to increase their girth. We also give a new perspective on LDPC protograph-based parity-check matrices by viewing them as rows of a parity-check matrix equal to the sum of certain permutation matrices and obtain an important connection between all protographs and those with variable nodes of degree 2. We also show that the results and methodology that we develop for the all-one protograph can be used and adapted to analyze the girth of the Tanner graph of any parity-check matrix and demonstrate how this can be done using a well-known irregular, multi-edge protograph specified by the NASA Consultative Committee for Space Data Systems (CCSDS). Throughout the paper, we exemplify our theoretical results with constructions of LDPC codes with Tanner graphs of any girth between 6 and 14 and give sufficient conditions for a multi-step lifted parity-check matrix to have girth between 14 and 22.