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Communications in computer and information science

Quasi-Cyclic LDPC Codes Constructed Based on Row-Column Constrained Matrices

Hengzhou Xu, Baoming Bai, Hai Jiang Zhu, Mengmeng Xu, Bo Zhang

Published Jan 1, 2018• 1 citation
Publisher
DOI
10.1007/978-981-10-7877-4_27

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This article presents a search algorithm for constructing regular and irregular quasi-cyclic LDPC codes. Code rows and columns are divided into j (column weight) and k (row-weight) or more groups respectively. Rows or columns in a group are connected to the same group and in order of appearance. Grouping of rows and columns forms sub-matrices. The sequential row-column connections order creates a cyclic structure in sub-matrices. The row-column constraint is observed to avoid four-cycles. The proposed algorithm is flexible compared to other methods. It obtains codes over a wide range of girths, rates and lengths. Bit-error rate simulations show that obtained codes have good performance with randomly searched codes performing better that sequentially searched codes.
This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly LDPC codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterizations of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite geometry LDPC codes using the results developed in first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping sets of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite geometry and finite field cyclic and quasi-cyclic LDPC codes with large minimum weights are shown to have no harmful trapping sets with size smaller than their minimum weights. Consequently, their performance error-floors are dominated by their minimum weights.
For a high-rate case, it is difficult to randomly construct good low-density parity-check (LDPC) codes of short and moderate lengths because their Tanner graphs are prone to have short cycles. Also, the existing high-rate quasi-cyclic (QC) LDPC codes can be constructed only for very restricted code parameters. In this paper, based on special classes of cyclic difference families, we propose a new construction method of high-rate regular QC LDPC codes having parity-check matrices consisting of a single row of circulants with column-weight 3 or 4. The proposed QC LDPC codes can be constructed for various code rates and lengths including the minimum achievable length for given column-weight and design rate under girth 6. It is observed that the parity-check matrices of the proposed QC LDPC codes have full rank for column-weight 3 and just one redundant row for column-weight 4. It is shown that the error correcting performance of the proposed QC LDPC codes of short and moderate lengths is almost the same as that of the existing ones through numerical analysis.