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IEEE Transactions on Communications

Constructions of Type-II QC-LDPC Codes With Girth Eight from Sidon Sequence

Guohua Zhang, Yulin Hu, Yi Fang, Juhua Wang

Published Feb 25, 2019• 22 citations
Publisher

Abstract

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.

DOI
10.1109/tcomm.2019.2901481

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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.
In this work, we consider type-II quasi-cyclic LDPC codes with girth eight from a Sidon sequence. We first derive the necessary and sufficient conditions guaranteeing girth-eight type-II QC-LDPC codes. By combining these conditions and the concept of a Sidon sequence, two classes of type-II codes are subsequently proposed with girth up to eight. We discuss the distance upper bounds of the two classes of codes and show that the second class provides a larger distance upper bound. In particular, to the best of our knowledge, the second class we proposed yields the first algebraic construction for girth-eight type-II codes with rates larger than a half and distance upper bounds exceeding twelve. Via simulations, we show that the girth-eight type-II codes from the second class significantly outperform the existing CDF-based girth-eight type-II codes, and that they perform better than or almost identically to the randomly generated girth-eight quadr. congr. codes.
Quasi-cyclic low-density parity-check (QC-LDPC) codes are known to have algebraic structure, which suits hardware implementation. However, low girth makes their performances worse than random structured LDPC codes. Based on independent row-column mapping sequence, this work proposed a generalized method of constructing QC-LDPC codes with girth eight, which enlarges the class of girth eight QC-LDPC codes to a fairly general series. Simulation results show that the bit error rate of the constructed codes is close to that of progressive edge growth (PEG) based QC-LDPC codes under iterative decoding.
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.
For any column weight J and any row weight L, a novel framework is proposed such that a girth-eight (J,L) quasi-cyclic low-density parity-check (QC-LDPC) code with any block length above a lower bound can be constructed via a simple inequality in terms of greatest common divisor (GCD). The main advantage is that the construction of a class of (J,L) girth-eight QC-LDPC codes is transformed into a rather simple task, searching for J integers satisfying the so-called GCD constraint for L. Combining the new method with masking matrices, a class of type-1 QC-LDPC codes is presented with girth at least eight. Simulation results show that the type-1 codes perform better than the random QC-LDPC codes and quadratic-congruence-based QC-LDPC codes for moderate block lengths and low code rates.
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.