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IEEE Communications Letters

Construction of quasi-cyclic LDPC codes from quadratic congruences

Chun-Ming Huang, Jen-Fa Huang, Chao-Chin Yang

Published Apr 1, 2008• 49 citations
Publisher

Abstract

In this paper, we proposed a novel method for constructing quasi-cyclic low-density parity-check (QC-LDPC) codes based on circulant permutation matrices via a simple quadratic congruential equation. The main advantage is that QC-LDPC codes with a variety of block lengths and rates can be easily constructed with no cycles of length four or less. Simulation results show that the proposed QC-LDPC codes perform slight better than the random regular LDPC codes for short to moderate block lengths and have almost the same performance as Sridara-Fuja-Tanner codes.

DOI
10.1109/lcomm.2008.080065

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In this paper, we proposed a novel method for constructing quasi-cyclic low-density parity-check (QC-LDPC) codes based on circulant permutation matrices via a simple quadratic congruential equation. The main advantage is that QC-LDPC codes with a variety of block lengths and rates can be easily constructed with no cycles of length four or less. Simulation results show that the proposed QC-LDPC codes perform slight better than the random regular LDPC codes for short to moderate block lengths and have almost the same performance as Sridara-Fuja-Tanner codes.
In this paper, we proposed a new method for constructing quasi-cyclic low-density parity-check (QC-LDPC) codes based on circulant permutation matrices via the one-coincidence sequences. The main advantage is that QC-LDPC codes with a variety of block lengths and rates can be easily constructed with no cycles of length four or less. Simulation results show that the proposed QC-LDPC codes perform slight better than the random regular LDPC codes for short to moderate block lengths and have almost the same performance as Sridara- Fuja-Tanner codes.
The short paper proposes a method for constructing regular quasi-cyclic (QC) LDPC codes based on circulant permutation matrices via simple prime field operations. The main advantage is that regular QC LDPC codes with a variety of block lengths and rates can be easily constructed which have no cycles of length four or less. Simulation results show that within only a maximum of ten decoding iterations of sum-product algorithm(SPA) the constructed regular codes of high rates have no error floor down to the bit-error rate of 10-7.
In this paper a new low complexity method for constructing binary quasi-cyclic low-density parity-check (QC-LDPC) codes is introduced. In the proposed method, each block-column of the parity check matrix H is made by a circulant matrix in a way that the associated Tanner graph is free of cycle four. Each circulant matrix in H is made by a generator column. The generator columns should be selected in a way that each associated circulant matrix and every two distinct circulant matrices are free of cycle four. The generator columns are made by row distance sets. An algorithm for generating distance sets and obtaining circulant matrices with columns of weight three is presented separately. Simplicity of construction and having a good flexible family of quasi cyclic LDPC codes both in rate and length are the main properties of the proposed method. The performance of the proposed codes is compared with that of the random-like and Array LDPC codes over an AWGN channel. Simulation results show that from the performance perspective, the constructed codes are competitive with random-like and Array LDPC codes.
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.
In this paper, we propose a novel method with low complexity for constructing quasi-cyclic LDPC codes. The algorithm chooses every circulant permutation matrix sequentially, and makes sure that the current circulant permutation matrix forms no cycles of length 4 and 6 with the existent matrices, finally a family of (3,L)(3,L)(3,L) QC-LDPC codes with girth at least 8 is obtained. The main complexity of the algorithm is checking cycles of length 6, the number of which is no more than L2L^2L2. Simulation results show that, when block length is 1008 and using the iterative belief propagation decoding, the LDPC code constructed by the proposed algorithm out performs the random code about 0.1dB at the bit-error-rate of $ 10^{ - 5}$.
Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.