Quasi-Cyclic LDPC Codes of Column-Weight Two Using a Search Algorithm
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Research Article Quasi-Cyclic LDPC Codes of Column-Weight Two Using a Search Algorithm Gabofetswe Malema and Michael Liebelt School of Electrical and Electronic Engineering, The University of Adelaide, North Terrace, Adelaide 5005, SA, Australia Received 16 February 2006; Revised 11 August 2006; Accepted 6 February 2007 Recommended by Richard Heusdens This article introduces a search algorithm for constructing quasi-cyclic LDPC codes of column-weight two. To obtain a submatrix structure, rows are divided into groups of equal sizes. Rows in a group are connected in their numerical order to obtain a cyclic structure. Two rows forming a column must be at a specified distance from each other to obtain a given girth. The search for rows satisfying the distance is done sequentially or randomly. Using the proposed algorithm regular and irregular column-weight-two codes are obtained over a wide range of girths, rates, and lengths. The algorithm, which has a complexity linear with respect to the number of rows, provides an easy and fast way to construct quasi-cyclic LDPC codes. Constructed codes show good bit-error rate performance with randomly shifted codes performing better than sequentially shifted ones. Copyright © 2007 G. Malema and M. Liebelt. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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INTRODUCTION
Gallager has shown that column-weight-two codes have minimum distance increasing logarithmically with code length, compared to a linear increase when the column-weight is at least three [1]. Despite the low increase in minimum distance, these codes have shown potential in some applications such as partial response channels [2, 3]. These codes also have less computation because columns have only two connections. Although LDPC code performance has been shown to be good, their hardware implementation still remains a challenge. This is mainly because of their large sizes and complex random (unstructured) row-column connections. Applications have power, area, latency, and cost constraints that LDPC encoders and decoders must meet. Structured codes have been developed to reduce hardware implementation complexity by constraining code construction. However, constraining row-column connections may reduce the maximum girth (smallest cycle) attainable [4]. It has been shown that increasing the girth or average girth of a code increases its decoding performance [5, 6]. The girth also determines the number of iterations before a message propagates back to its original node. Performance of structured codes could therefore be improved by increasing their girths.
In [7] girth 16 and 18 codes with row-weights of 4 and 3 are constructed from graphical models. The method does not provide an easy way of constructing codes for high row-weights and expanding codes. In [3] cyclic codes are constructed algebraically with girth of twelve for rowweights of k, where k − 1 i
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