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Electronics Letters

Type‐II quasi‐cyclic low‐density parity‐check codes from Sidon sequences

Guohua Zhang

Published Feb 8, 2016• 13 citations
Publisher

Abstract

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.

DOI
10.1049/el.2015.2634

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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.
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.
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.
For any row weight L , a novel class of (8, L ) quasi‐cyclic low‐density parity‐check (QC‐LDPC) codes is deterministically constructed with girth eight. Compared with a similar construction, the proposed (7, L ) and (8, L ) QC‐LDPC codes exist for any circulant permutation matrix of size P ≥ ( L − 1)( L 2 + 3 L /4 + 3/4) + 1 and P ≥ ( L − 1)( L 2 + 3 L /4 + 7/4) +1, which remarkably improve the existing bounds of ( L − 1)( L 2 + L ) +1 and ( L − 1)( L 2 + L + 1) +1, respectively. Moreover, for any column weight J , any row weight L and any common difference, a construction for ( J , L ) QC‐LDPC codes with girth eight is also proposed. This construction has high flexibility with respect to the design of code rate and code length. Simulation results show that the novel codes with moderate lengths perform well in the additive white Gaussian noise channel.
Multi-type quasi-cyclic (QC) low-density parity-check (LDPC) codes can be considered as multiple-edge protograph QC-LDPC codes having some advantages in the minimum Hamming distance bound over single-edge protograph codes or type-I QC-LDPC codes when the base matrices have the same size. In this paper, we investigate a class of multi-type QC-LDPC codes whose parity-check matrices contain just one blockrow of circulants and we obtain the generator matrix of such codes in general form. Using the permutation arrays and defining injection arrays, we present a new approach to construct a class of high-rate type-I QC-LDPC codes with girth 6 from the constructed 4-cycle free multi-type QC-LDPC codes. In continue, for 2 ≤ w ≤ 6, some type-w QC-LDPC codes with girth 6 are constructed explicitly such that the constructed codes are flexible in terms of rate and length. To the best of our knowledge, for w = 5,6, this is the first paper which deals with the explicit construction of type-w QC-LDPC codes with girth 6 and high rates. Moreover, for w = 3,4, the constructed type-w QC-LDPC codes have better (6,8)-cycle multiplicities than the codes with minimum achievable length recently constructed by cyclic difference families (CDFs). Simulation results show that the binary and non-binary constructed codes outperform the constituent underlying QC-LDPC 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.
For any row weight L and any circulant permutation matrix size P > ( L − 1) 2 + b 0 ( L − 1) ( b 0 = L − 2 + mod( L ,2)), a class of (3, L ) quasi‐cyclic (QC) low‐density parity‐check (LDPC) codes is explicitly constructed with girth eight, based on the quadratic function f ( x ) = x 2 + bx and its derivative, where b is either of the two integers { b 0 , − b 0 − 2( L − 2)}. Compared with a similar construction from the so‐called difference sequence, the proposed construction is not only much more simple in concept, but also completely deterministic in the sense that no computer search or computing is required. Simulation results show that the new codes significantly outperform the girth‐eight codes constructed by the earliest‐sequence‐based method.