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IEEE Transactions on Communications

Construction of Low-Density Parity-Check Codes by Superposition

J. Xu, Lei Chen, Lingqi Zeng, Longwen Lan, Shu Lin

Published Feb 1, 2005• 88 citations
Publisher

Abstract

This paper presents a superposition method for constructing low-density parity-check (LDPC) codes. Several classes of structured LDPC codes are constructed. Codes in these classes perform well with iterative decoding, and their Tanner graphs have girth at least six.
DOI
10.1109/tcomm.2004.841966

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Low-density parity-check codes (LDPC) can have an impressive performance under iterative decoding algorithms. In this paper we introduce a method to construct high coding gain lattices with low decoding complexity based on LDPC codes. To construct such lattices we apply Construction D', due to Bos, Conway, and Sloane, to a set of parity checks defining a family of nested LDPC codes. For the decoding algorithm, we generalize the application of max-sum algorithm to the Tanner graph of lattices. Bounds on the decoding complexity are derived and our analysis shows that using LDPC codes results in low decoding complexity for the proposed lattices. The progressive edge growth (PEG) algorithm is then extended to construct a class of nested regular LDPC codes which are in turn used to generate low density parity check lattices. Using this approach, a class of two-level lattices is constructed. The performance of this class improves when the dimension increases and is within 3 dB of the Shannon limit for error probabilities of about 10-6. This is while the decoding complexity is still quite manageable even for dimensions of a few thousands
In this paper, we propose a novel and simple construction of regular LDPC codes using sparse binary sequences. The distribution matrix of ones in a parity-check matrix is introduced into this new construction. The cyclic cross correlation function and the cyclic auto correlation function are employed to ensure that the girth of Tanner graph is at least eight. The new codes perform well using the sum-product algorithm for iterative decoding. Low encoding complexity can also be achieved due to the inherent quasi-cyclic structure of the LDPC codes.
A novel and simple construction of regular LDPC codes was proposed, which used sparse binary sequences, through studying construction methods of LDPC codes. The distribution matrix of ones in parity-check matrix was introduced into the new construction. The cyclic cross correlation function and the cyclic auto correlation function were utilized to ensure that the girth of Tanner graph was eight. The new codes perform well using the sum-product algorithm for iterative decoding. Low encoding complexity can also be achieved due to the inherent quasi-cyclic structure of the LDPC codes.
Two algebraic methods for systematic construction of structured regular and irregular low-density parity-check (LDPC) codes with girth of at least six and good minimum distances are presented. These two methods are based on geometry decomposition and a masking technique. Numerical results show that the codes constructed by these methods perform close to the Shannon limit and as well as random-like LDPC codes. Furthermore, they have low error floors and their iterative decoding converges very fast. The masking technique greatly simplifies the random-like construction of irregular LDPC codes designed on the basis of the degree distributions of their code graphs