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IEEE Communications Letters

Construction of QC-LDPC Codes from Sidon Sequence Using Permutation and Segmentation

Lintao Zhang, Juhua Wang

Published May 23, 2022• 9 citations
Publisher

Abstract

By utilizing a Sidon sequence, two new methods are proposed to construct quasi-cyclic (QC) low-density parity-check (LDPC) codes. The first method yields a QC-LDPC code without 4-cycles and the second method gives a QC-LDPC code without 4-cycles and 6-cycles. In the literature there are two methods which employ mathematical objects somewhat resembling a Sidon sequence to remove 4-cycles, and there are two explicit constructions which guarantee the absence of 4-cycles and 6-cycles. Compared with the former, our first method is much more flexible in selecting parameters while providing identical or better performance. Compared with the latter, our second method provides much better performance.

DOI
10.1109/lcomm.2022.3177511

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By utilizing a Sidon sequence, two new methods are proposed to construct quasi-cyclic (QC) low-density parity-check (LDPC) codes. The first method yields a QC-LDPC code without 4-cycles and the second method gives a QC-LDPC code without 4-cycles and 6-cycles. In the literature there are two methods which employ mathematical objects somewhat resembling a Sidon sequence to remove 4-cycles, and there are two explicit constructions which guarantee the absence of 4-cycles and 6-cycles. Compared with the former, our first method is much more flexible in selecting parameters while providing identical or better performance. Compared with the latter, our second method provides much better performance.
From Sidon sequences, two explicit methods for constructing (2 J , 2 L )‐regular type‐II quasi‐cyclic (QC) low‐density parity‐check (LDPC) codes are proposed without four cycles. The advantage of the new methods is two‐fold: (i) compared with the existing lower bound of circulant permutation matrix (CPM) sizes above which type‐II QC‐LDPC codes exist without four cycles, the novel constructions enable much tighter lower bounds; and (ii) the codes constructed by the first method perform almost as well as type‐II QC‐LDPC codes from perfect cyclic difference set, while possessing a much more flexible CPM sizes.
In this paper, we consider the constructions of type-II quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight from a Sidon sequence. We first derive the necessary and sufficient conditions guaranteeing a girth-eight type-II QC-LDPC code. By combining these conditions with the concept of the Sidon sequence, three classes of type-II QC-LDPC codes are subsequently proposed with girth eight. To the best of our knowledge, the second and the third classes we proposed are the first series of systematic and algebraic constructions for the girth-eight type-II QC-LDPC codes that have rates being greater than a half and, at the same time, have distance upper bounds being not limited to 12. Via simulations, we show the promising performance of the proposed type-II QC-LDPC codes with girth eight. In addition, a set of general bounds and explicit/random constructions without using a Sidon sequence are also presented for the type-II QC-LDPC codes with girth six or eight. We observe that the proposed codes perform better than or equally well as the random QC-LDPC codes from PEG and quadr.-congr. methods, while the novel codes possess both highly structured parity-check matrices and very flexible choices in circulant size.
Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes have practical significance in the field of engineering, thus their construction algorithm is a hot issue in the field of LDPC codes. Based on existing construction methods, particularly the method based on Progressive Edge-Growth (PEG) algorithm, a new construction method of shift matrix is proposed. It can effectively avoid the copy of cycle structures, especially short cycles in base matrix after extension. Besides, it also relieves the time cost caused by random search, and greatly improves the length of cycles when compared with algorithms based on methods such as quadratic congruence and arithmetic sequence. Those algorithms can only eliminate short cycles with length 4, while our method can provide QC-LDPC codes without length 6 at least. Cycle analysis and simulation results show that the newly constructed QC-LDPC codes have good cycle structure and error correction performance.
Low-density Parity-check (LDPC) coding is a class of advanced channel coding techniques. To improve the performance of LDPC codes, a simple and effective method to construct QC-LDPC codes without short cycles (cycles of length four and six) is proposed. The approach is especially effective when the parity-check matrix of QC-LDPC codes is composed of circulant permutation matrices. Firstly, we analyze the shapes of the cycles in the exponent matrix, to get the required conditions of no four-cycles and six-cycles. Then we show how to construct the parity-check matrix without four-cycles and six-cycles by filling rows of the exponent matrix with arithmetic sequences. Simulation results show that the proposed codes have better performance than array codes. Though the performance of the proposed codes is not as well as random codes at higher Eb/N0, they can be applied to practical communications because of being encoded in linear time with shift registers.
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.