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IEEE Information Theory Workshop (ITW)

Type-II Quasi-Cyclic LDPC Codes with Girth Eight from Sidon Sequence

Guohua Zhang, Yulin Hu, Qinwei He, Juhua Wang

Published Nov 1, 2018• 1 citation
Publisher

Abstract

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.

DOI
10.1109/itw.2018.8613426

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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.
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.
From Sidon sequence over ZP, two classes of type-1 (J = 3;L) QC-LDPC codes are explicitly proposed with block length L2P and with girth at least ten. For the first method, any Sidon sequence over ZP(P odd) with cardinality no smaller than L (L odd) corresponds to a class of type-1 (3, L) QC-LDPC codes with girth at least ten. For the second one, any Sidon sequence over ZP(P prime) with cardinality no smaller than L (L arbitrary) corresponds to a class of type-1 (3, L) QC-LDPC codes with girth at least ten. Compared with the method from 3-D lattices, which can also generate type-1 QC-LDPC codes with girth at least ten, the main advantage of the new methods is that the construction process is much simpler, and L is not necessarily a prime integer. Simulation results show that the new type-1 QC-LDPC codes outperform the girth-8 QC-LDPC codes constructed by Sidon sequence or the earliest sequence, and perform almost as well as Bocharova's shortest girth-12 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.
Type-II quasi-cyclic (QC) LDPC codes are constructed from combinations of weight-0, weight-1 and weight-2 circulant matrices. The structure of cycles of length 2nare investigated, and necessary and sufficient conditions for a type-II QC LDPC parity check matrix H to have girth at least 2(n+ 1) are given. An explicit construction of type-II codes which guarantees girth at least 6 is presented. A necessary and sufficient condition for a QC matrix with one or more rows of circulants, to be fullrank is derived.
For any row weight L, several novel classes of (J = 5, L) and (J = 6, L) quasi-cyclic LDPC codes are deterministically constructed with girth eight. From these results, it is proved that (5, L) QC-LDPC codes with girth eight exist for any circulant permutation matrix (CPM) size P ≥ (2L + 3)(L - 1) + 1, and that girth-eight (6, L) QC-LDPC codes exist for any P ≥ 2(L+5)(L-1)+1. The two novel bounds remarkably improve the existing bounds of L2(L-1) + 1 and (L2+1)(L-1)+1, respectively. Moreover, for any column weight J and any row weight L, a construction for (J,L) QC-LDPC codes with girth eight is also proposed. This is the first deterministic and systematic construction which can generate girth-eight QC-LDPC codes with J ≥ 7.