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World Congress on Information and Communication Technologies

Iterative threshold decoding of Quasi-Cyclic One Step Majority logic decodable codes

Karim Rkizat, Mohammed Lahmer, Mostafa Belkasmi

Published Dec 1, 2015• 3 citations
Publisher

Abstract

This paper presents a new class of Quasi-Cyclic One Step Majority logic codes of 1/2 rate constructed from perfect difference set. Theses codes can be encoded with low complexity, and perform very well when decoded with the Iterative threshold decoding algorithm. Much of these codes is a subfamily of the LDPC codes and can be decoded using belief propagation algorithm. A comparison between our results and those for LDPC code in terms of BER performance are presented.

DOI
10.1109/wict.2015.7489656

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This paper presents a new class of Quasi-Cyclic One Step Majority logic codes of 1/2 rate constructed from perfect difference set. Theses codes can be encoded with low complexity, and perform very well when decoded with the Iterative threshold decoding algorithm. Much of these codes is a subfamily of the LDPC codes and can be decoded using belief propagation algorithm. A comparison between our results and those for LDPC code in terms of BER performance are presented.
This paper investigates the construction and iterative threshold decoding of low rate Quasi-Cyclic One Step Majority logic codes based on combinatorial designs. These codes are constructed with two type of difference family: cyclic disjoint difference sets and cyclic disjoint difference family. They can be encoded with low complexity, and perform very well when decoded with the Iterative threshold decoding (ITD) algorithm. We present an amelioration of ITD algorithm for decoding these codes. This algorithm modification can be applicable for every code rate. The performances of this algorithm for decoding these codes on Additive White Gaussian Noise (AWGN) channel, is investigated.
Among the classes of LDPC codes that have been constructed and designed, the only class of LDPC codes that are cyclic is the class of codes constructed based on the incidence vectors of lines of finite geometries, called finite geometry LDPC codes. Cyclic finite geometry LDPC codes are known to have large minimum distances and can provide good error performance with very low error-floors using iterative decoding based on belief propagation. Their cyclic structure allows them to be efficiently and systematically encoded with simple shift-registers in linear time with linear complexity. An obvious question is whether, besides cyclic finite geometry LDPC codes, there are other cyclic codes with large minimum distances that can be efficiently decoded iteratively using channel soft information. In this paper, we present one such class of cyclic codes. Codes in this class are two-step majority-logic decodable and they are also constructed based on finite geometries. Two iterative decoding algorithms are devised for this class of cyclic codes and they provide significant coding gain over the two-step majority-logic decoding of codes in this class.
Codes constructed based on finite geometries form a large class of cyclic codes with large minimum distances which can be decoded with simple majority-logic decoding in one or multiple steps. In 2001, Kou, Lin and Fossorier showed that the one-step majority-logic decodable finite geometry codes form a class of cyclic LDPC codes whose Tanner graphs are free of cycles of length 4. These cyclic finite geometry LDPC codes perform very well over the AWGN channel using iterative decoding based on belief propagation (IDBP) and have very low error-floors. However, the standard IDBP is not effective for decoding other cyclic finite geometry codes because their Tanner graphs contain too many short cycles of length 4 which severely degrade the decoding performance. This paper investigates iterative decoding of two-step majority-logic decodable finite geometry codes. Three effective algorithms for decoding these codes are proposed. These algorithms are devised based on the orthogonal structure of the parity-check matrices of the codes to avoid or reduce the degrading effect of the short cycles of length 4. These decoding algorithms provide significant coding gains over the standard IDBP using either the sum-product or the min-sum algorithms.
Previously, the belief propagation (BP) algorithm has received a lot of attention in the coding community, mostly due to its near-optimum decoding for low-density parity check (LDPC) codes and its connection to turbo decoding. In this paper, we investigate the performance achieved by the BP algorithm for decoding one-step majority logic decodable (OSMLD) codes. The BP algorithm is expressed in terms of likelihood ratios rather than probabilities, as conventionally presented. The proposed algorithm fits better the decoding of OSMLD codes with respect to its numerical stability due to the fact that the weights of their check sums are often much higher than that of the corresponding LDPC codes. Although it has been believed that OSMLD codes are far inferior to LDPC codes, we show that for medium code lengths (say between 200-1000 bits), the BP decoding of OSMLD codes can significantly outperform BP decoding of their equivalent LDPC codes. The reasons for this behavior are elaborated.
In this paper, we devise a non-binary version of the Soft-Input Soft-Output Iterative Threshold Decoder (ITD). This is a continuation of our work where the devised algorithm is the upgrade of the non-iterative non-binary Soft-Input Hard-Output Threshold Decoder (TD), which is itself an upgrade of previous work on the Hard-Input Hard-Output non-binary Majority-Logic Decoder (MLGD). The proposed ITD showed an improvement compared to its predecessors where coding gains of 1.5dB and 2.5dB were observed compared to the TD algorithm and the MLGD algorithm respectively, at a bit error rate (BER) of 10-5. The proposed decoder is best suited for One-Step Majority-Logic decodable codes with a large orthogonal structure in their dual codes, such as finite geometry codes, which are also Low-Density Parity-Check codes. The performances of our decoder are presented in terms of bit, symbol, and block error rates, and the proposed decoder has shown to outperform the best well-known decoders for LDPC codes as displayed by the simulation results. Where the proposed decoder offers a coding gain of 1dB compared to the Min-Max algorithm, 1.5dB of coding gain compared to the ISRB algorithm, and 1.6dB of coding gain compared to the KV algorithm at a block error rate (BLER) of 10-4. Also, the computational complexity of our decoder is proven to be lower than that of the ISRB algorithm, which has itself been proven to be less complex than the other algorithms in the study conducted by its authors.
A reliability-based iterative majority-logic decoding algorithm for regular low-density parity-check (LDPC) codes was recently proposed by Huang et al. In this paper we present an improved version of that algorithm by introducing a different reliability measure for each check-sum of the parity-check matrix, and taking it into account in the computation of the extrinsic information that is used to update the reliability measure of each received bit in each iteration. Some simulations results are given, which show that the new algorithm, while requiring very little additional computational complexity, not only achieves a considerable error performance gain over the standard one, but also, importantly, outperforms the iterative decoding based on belief propagation (IDBP), especially for short and medium block length finite-geometry (FG) LDPC codes.