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International Conference on Advanced Communication Systems and Information Security (ACOSIS)

Construction and iterative threshold decoding for low rates Quasi-Cyclic One Step Majority logic decodable codes

Karim Rkizat, Mohammed Lahmer, Mostafa Belkasmi, Saïd Nouh

Published Oct 1, 2016• 1 citation
Publisher

Abstract

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.

DOI
10.1109/acosis.2016.7843944

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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.
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.
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.
This correspondence presents a construction of quasicyclic (QC) low-density parity-check (LDPC) codes based on a special type of combinatorial designs known as block disjoint difference families (BDDFs). The proposed construction of QC-LDPC codes gives parity-check matrices with column weight three and Tanner graphs having a girth lower-bounded by 6. The proposed QC-LDPC codes provide an excellent performance with iterative decoding over an additive white Gaussian-noise (AWGN) channel. Performance analysis shows that the proposed short and moderate length QC-LDPC codes perform as well as their competitors in the lower signal-to-noise ratio (SNR) region but outperform in the higher SNR region. Also, the codes constructed are quasicyclic in nature, so the encoding can be done with simple shift-register circuits with linear complexity.
In the late 1950s and early 1960s, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for hard-decision algebraic decoding, such as Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes. This paper shows that finite fields can also be successfully used to construct algebraic low-density parity-check (LDPC) codes for iterative soft-decision decoding. Methods of construction are presented. LDPC codes constructed by these methods are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN), binary random, and burst erasure channels with iterative decoding in terms of bit-error probability, block-error probability, error-floor, and rate of decoding convergence, collectively. Particularly, they have low error floors. Since the codes are QC, they can be encoded using simple shift registers with linear complexity.