To provide reliable transmissions with flexible rates for satellite communications, this paper presents a novel method of constructing multi-rate quasi-cyclic low-density parity-check (LDPC) codes. The basic idea is to generate low-rate codes from a high-rate mother code by combining shortening and extending, which ensures that the generated code family owns the same code length, in order to maintain the same frame structure. The code construction involves the design of base matrices and exponent matrices for the designed codes. A progressive row elimination and addition algorithm is proposed for designing the code base matrices from a high rate to low rates. This algorithm leads to the nested and systematic structure of the parity-check matrices, which are desirable for practical implementations of their encoders and decoders, while ensuring the optimal decoding thresholds. In addition, we construct a circulation coefficient matrix based on finite fields and select the optimal rows in this matrix to construct exponent matrices while considering of cycle structures. We demonstrate that the designed codes achieve better performance for all the code rates than the LDPC codes in DVB-S2X standards. In addition, the proposed codes do not exhibit error floors for their block error rates down to 10 ⁻⁵ .

This paper presents a new type of globally-coupled low-density parity-check (GC-LDPC) codes whose base matrix has a cyclic structure in the global part. Therefore, the resulting codes are referred to as tail-biting GC-LDPC (TB-GC-LDPC) codes. We propose two methods to construct TB GC quasi-cyclic LDPC (TB-GC-QC-LDPC) codes. For the first method, we extract a replicated version of a constructed base matrix and mask it with a designed masking matrix. Compared to the conventional construction methods, this method provides more flexibility in code length for TB-GC-QC-LDPC codes. The second method is based on designing the incidence matrix of a special type of packings. Examples show that the constructed TB-GC-QC-LDPC codes perform well over the additive white Gaussian noise channel (AWGNC) and the binary erasure channel (BEC). The asymptotic performance of TB-GC-LDPC ensembles over BECs are also analyzed by resorting to density evolution. Moreover, numerical results show that TB-GC-LDPC ensembles can achieve better flooding-schedule decoding (FSD) thresholds than the corresponding GC-LDPC ensembles with a similar structure. With sufficient decoding iterations in the global phase, the local/global two-phase iterative decoding (TPD) thresholds of the TB-GC-LDPC ensembles significantly outperform those of the corresponding GC-LDPC ensembles as well.

In this letter, we present an efficient method for constructing a family of quasi-cyclic low-density parity-check (QC-LDPC) codes specified only by one single exponent matrix that can support multiple lifting sizes. Our method adopts the small-lifting-size-first design strategy and the resulting exponent matrix of larger lifting size has non-decreasing girth. Numerical results show that all constituent codes can perform as well as (outperform) the CCSDS (5G) LDPC codes over the AWGN channel with the iterative decoding algorithms. This fact demonstrates the feasibility and validity of our proposed method and makes it competitive in some applications, e.g., enhancing short and medium-length LDPC codes to support longer length.

M. Fossorier proposed how to determine the necessary and sufficient conditions for the existence of cycles in the Tanner graph of quasi‐cyclic LDPC (QC‐LDPC) codes, which has been widely investigated in the study of LDPC codes. This paper presents some new necessary and sufficient conditions for the existence of cycles with arbitrary lengths and proposes a simple and novel method for counting cycles of QC‐LDPC codes based on the improved conditions. Numerical results show that, compared with the existing methods, the presented method is effective and feasible and can enumerate cycles of QC‐LDPC codes in a cyclic‐shift way.

This paper presents a family of rate‐compatible (RC) globally coupled low‐density parity‐check (GC‐LDPC) codes, which is constructed by combining algebraic construction method and graph extension. Specifically, the highest rate code is constructed using the algebraic method and the codes of lower rates are formed by successively extending the graph of the higher rate codes. The proposed rate‐compatible codes provide more flexibility in code rate and guarantee the structural property of algebraic construction. It is confirmed, by numerical simulations over the AWGN channel, that the proposed codes have better performances than their counterpart GC‐LDPC codes formed by the classical method and exhibit an approximately uniform gap to the capacity over a wide range of rates. Furthermore, a modified two-phase local/global iterative decoding scheme for GC‐LDPC codes is proposed. Numerical results show that the proposed decoding scheme can reduce the unnecessary cost of local decoder at low and moderate SNRs, without any increase in the number of decoding iterations in the global decoder at high SNRs.

This paper studies on a class of quasi-cyclic LDPC (QC-LDPC) codes, i.e., Tanner (3, 23)-regular QC-LDPC codes of code length 23 p with p being a prime and p ≡ 1 (mod 69). We first analyze the cycle structure of Tanner (3, 23)-regular QC-LDPC codes, and divide their cycles of lengths 4, 6, 8, and 10 into five equivalent types. We propose the sufficient and necessary condition for the existence of these five types of cycles, i.e., the polynomial equations in a 69th unit root of the prime field F _p . We check the existence of solutions for such polynomial equations by using the Euclidean division algorithm and obtain the candidate girth values of Tanner (3, 23)-regular QC-LDPC codes. We summarize the results and determine the girth distribution of Tanner (3, 23)-regular QC-LDPC codes.