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IEEE/CIC International Conference on Communications in China (ICCC)

On Disjoint Difference Sets and Their Associated QC-LDPC Codes with Large Girth

Guohua Zhang, Mengdi Ni, Yi Fang

Published Aug 10, 2023• 2 citations
Publisher

Abstract

The disjoint difference set (DDS) has been utilized in recent years to construct quasi-cyclic (QC) low-density parity-check (LDPC) codes without small cycles. However, the existing DDS-based QC-LDPC codes have no all-zero circulants, which limits the decoding performance of such codes to some extent. In this paper, two new types of DDS-based QC-LDPC codes are proposed, which contain all-zero circulants and possess girth at least eight and girth at least ten, respectively. Numerical simulations show that the new codes have good performance in AWGN channels. Besides, a new result on existence for DDSs and two novel deterministic constructions for DDSs are put forward.

DOI
10.1109/icccworkshops57813.2023.10233824

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The disjoint difference set (DDS) has been utilized in recent years to construct quasi-cyclic (QC) low-density parity-check (LDPC) codes without small cycles. However, the existing DDS-based QC-LDPC codes have no all-zero circulants, which limits the decoding performance of such codes to some extent. In this paper, two new types of DDS-based QC-LDPC codes are proposed, which contain all-zero circulants and possess girth at least eight and girth at least ten, respectively. Numerical simulations show that the new codes have good performance in AWGN channels. Besides, a new result on existence for DDSs and two novel deterministic constructions for DDSs are put forward.
A novel relationship between disjoint difference sets (DDS) and QC-LDPC codes without cycles of length up to six is established. Based on DDS with two sets, a class of column-weight-four QC-LDPC codes is constructed with girth at least eight. In the literature there are several explicit or semi-random methods which can yield column-weight-four QC-LDPC codes without 4-cycles and 6-cycles. Compared with these existing methods, the new DDS-based method is able to offer QC-LDPC codes with desirable decoding performance and flexible circulant sizes, and at the same time able to provide a large number of non-equivalent candidate codes. Moreover, by employing DDS with only one set, a class of column-weight-three QC-LDPC codes is constructed with short lengths and with girth at least eight.
This paper presents new half rate Quasi Cyclic Low Density Parity Check (QC- LDPC) codes formed on the basis of combinatorial designs. In these codes, circulant matrices of the parity check matrix are formed on the basis of subsets in which the difference between any two elements of a subset is unique with all differences obtained from the same or different subsets. This structure of circulant matrices guarantees non-existence of cycle-4 in the Tanner graph of QC-LDPC codes. First, an irregular code with girth 6 constituted by two rows of circulant matrices is proposed. Then, more criteria will be considered on the structure of subsets with the mentioned feature aiming to represent a new scheme of regular QC-LPDC codes with girth at least 8. From simulations, it is confirmed that codes have similar to or better performance than other well-known half rate codes, while require lower complexity in their design.
A novel class of quasi-cyclic (QC) low-density parity-check (LDPC) codes is explicitly proposed without 4-cycles and 6-cycles via the greatest-common-divisor (GCD) framework. From the new codes, a novel type of girth-eight QC-LDPC codes with a dual-diagonal (DD) structure is presented, in which the single undetermined entry in the associated exponent matrix can be determined by simple formulae. The proposed DD codes have similar or better performance compared with DD codes obtained from computer search. Moreover, a lower bound on circulant sizes which guarantees the existence of qualified undetermined entries is also provided.
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.
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.