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IEEE International Symposium on Information Theory (ISIT)

A class of quasi-cyclic regular LDPC codes from cyclic difference families with girth 8

Masaya Fujisawa, Shojiro Sakata

Published Jan 1, 2005• 6 citations
Publisher

Abstract

In this paper, we propose a class of regular LDPC codes from a cyclic difference family, which is a kind of combinatorial design. These LDPC codes have no 4-cycles, i.e., cycles of length 4. We clarify the conditions on which these codes with column weight 3 have no 6-cycles and discuss their minimum distance. Finally, we show the performance of the proposed codes with high rates and moderate lengths

DOI
10.1109/isit.2005.1523756

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In this paper, we propose a class of regular LDPC codes from a cyclic difference family, which is a kind of combinatorial design. These LDPC codes have no 4-cycles, i.e., cycles of length 4. We clarify the conditions on which these codes with column weight 3 have no 6-cycles and discuss their minimum distance. Finally, we show the performance of the proposed codes with high rates and moderate lengths
In this paper we propose a method of constructing quasi-cyclic regular LDPC codes from a cyclic difference family, which is a kind of combinatorial design. The resulting codes have no 4-cycle, i.e. cycles of length four and are defined by a small set of generators of codes with high rate and large code length. In particular, for LDPC codes with column weight three, we clarify the conditions on which they have no 6-cycle and their minimum distances are improved. Finally, we show the performance of the proposed codes with high rates and moderate lengths.
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.
This paper presents a novel regular Quasi-Cyclic (QC) Low Density Parity Check (LDPC) codes with column-weight three and girth at least eight. These are designed on the basis of combinatorial design in which subsets applied for the construction of circulant matrices are determined by a particular subset. Considering the non-existence of cycles four and six in the structure of the parity check matrix, a bound for their minimum weight is proposed. The simulations conducted confirm that without applying a masking technique, the newly implemented codes have a performance similar to or better than other well-known codes. This is evident in the waterfall region, while their error floor at very low Bit Error Rate (BER) is expected.
This paper is devoted to introduce a special classes of (QC-LDPC) with very restricted code parameters based on planar difference families. Such difference families could be obtained by numerical analysis and computer programs. The resulting codes have parity check matrices with column-weight greater than three, at least no 4-cycle and approximately full rank. It can be noted that the construction based on planar difference families exhibits more flexibility than that based on difference sets in terms of length and code rate selections. Besides, the more increasing in the column-weights of parity check matrices of QC-LDPC codes, the more improvement in the minimum distances of them. Simulation results show that over the additive white Gaussian noise channel, these codes could outperform their randomly constructed counterparts.
A novel relationship between disjoint difference sets (DDS) and QC-LDPC codes without cycles of length up to six is established. Based on DDS with two sets, a class of column-weight-four QC-LDPC codes is constructed with girth at least eight. In the literature there are several explicit or semi-random methods which can yield column-weight-four QC-LDPC codes without 4-cycles and 6-cycles. Compared with these existing methods, the new DDS-based method is able to offer QC-LDPC codes with desirable decoding performance and flexible circulant sizes, and at the same time able to provide a large number of non-equivalent candidate codes. Moreover, by employing DDS with only one set, a class of column-weight-three QC-LDPC codes is constructed with short lengths and with girth at least eight.
This paper,based on algebra and graph,proposes a method for constructing structured low-density parity-check(LDPC) codes.In this method,the authors design a connected graph with three kinds of special path to ensure that the Tanner graph of the parity check matrix mapped from the connected graph is without short cycles.The construction method results in two classes of(3,k)-regular quasi-cyclic LDPC codes with girth 8 and 12 respectively.Furthermore,by extending this construction method,two classes of LDPC codes with column weight 2,whose girths are respectively 16 and 24,can be obtained.The simulation studies show that these codes can achieve better performance than randomly constructed regular LDPC codes over AWGN channels with iterative decoding.At the BER of 10-5,the new proposed(3,6)-regular(4356,2205) QC-LDPC code performs 1.48dB from the Shannon limit.Moreover,it outperforms the random regular LDPC code of comparable parameters by 0.1dB.For the new proposed structured LDPC codes with column weight 2,they can achieve about 2dB coding gains with respect to the random regular LDPC codes with similar block lengths and code rates.