The performance of quasi-cyclic low-density parity-check (QC-LDPC) codes is adversely impacted by the presence of short cycles, particularly 4-cycles and 6-cycles. Existing methods to eliminate these short cycles can be generally divided into two categories: search-based methods and explicit methods. Search-based methods, such as symmetrical construction, typically involve extensive searches for exponent matrices that satisfy specific structural constraints. This results in a complex search process and high description complexity for QC-LDPC codes. In contrast, explicit methods, such as the greatest common divisor (GCD) and shifting sequence (SS) methods, leverage specific mathematical formulas to directly define the required exponent matrix. This eliminates the need for computer-based searches and leads to lower description complexity. Additionally, the column weight is a crucial factor influencing the performance of QC-LDPC codes. Existing explicit construction methods that ensure the absence of 4-cycles and 6-cycles are primarily applicable to QC-LDPC codes with column weight 3. However, methods suitable for column weight 4 are still relatively rare. The objective of this study is to investigate a novel explicit construction method for QC-LDPC codes with column weight 4, girth 8, and excellent decoding performance.

Inspired by the SS method for constructing QC-LDPC codes with column weight 3, this paper introduces a novel SS method to design QC-LDPC codes with column weight 4. The core methodology involves two steps: directly defining two original sequences using mathematical formulas and then generating two derived sequences by right shifting the original sequences. These four sequences together form the required exponent matrix. Subsequently, an analysis is conducted to verify that for any circulant size above a certain lower bound, the governing equations for 4-cycles and 6-cycles do not hold. Finally, the sum-product algorithm (SPA) is employed to simulate the newly constructed codes and compare their decoding performances with those of several typical QC-LDPC codes.

Through a rigorous mathematical analysis of the governing equations for 4-cycles and 6-cycles, it is concluded that when the circulant size is greater than or equal to the difference between the maximum element and half of the row weight, neither 4-cycles nor 6-cycles exist. Moreover, due to the presence of 8-cycles, the newly constructed codes exhibit a girth of exactly 8. Simulation results indicate that the new codes outperform codes constructed using the GCD method and perform similarly to codes based on symmetrical construction but with a significantly simpler construction process. Furthermore, the new codes have the potential to outperform 5G codes in the high-signal-to-noise region.

A novel explicit construction method for high-performance QC-LDPC codes is proposed, offering the following advantages: 1) applicability to any row weight L and 2) elimination of the need for exhaustive searches, allowing QC-LDPC codes to be explicitly constructed through mathematical formulas, thereby significantly reducing description complexity.

Indoor visible light communication (VLC) suffers from the low modulation bandwidth of a single light-emitting diode (LED), which hardly meets the demand for high-rate transmission. As a multiple-input-multiple-output (MIMO) technology, generalized spatial modulation (GSM) allows indoor VLC systems to achieve high spectral efficiency and excellent anti-interference capability using multiple LEDs for data transmission. However, in indoor MIMO-GSM-VLC systems, narrow spacing between multiple LEDs makes the channel characteristics between different LEDs and photodetectors very similar, resulting in a high correlation of the channel matrix. Because of the special characteristics of the VLC channel, the existing constellation mapping and channel coding schemes are no longer feasible for indoor MIMO-GSM-VLC systems. Therefore, incorporating the spatial characteristics of the VLC channel to develop a high-reliability and high-efficiency optical transmission scheme is imperative.

The proposed optical transmission scheme is divided into two parts: a constellation mapping scheme and an improved protograph low-density parity-check (LDPC) code. First, this paper proposes a novel spatial multipulse position modulation (MPPM) mapping scheme, called the unequal power spatial MPPM (UPSM) constellation, to realize the joint optimization of GSM and MPPM. In particular, owing to the correlation of the VLC channel matrix, effective LED activation groups with similar characteristics transmitting MPPM symbols will cause serious intergroup interference in the MIMO-GSM-VLC system. Based on the VLC channel matrix, the principle and calculation method of influence coefficients for LED activation groups are presented to reallocate the peak transmit power of the MPPM symbols for effective LED activation groups, which can efficiently mitigate the attenuation of MPPM symbols in transmission. In addition, the maximum Hamming distance principle is used to optimize the mapping relationship between MPPM labels and MPPM symbols in the UPSM constellation. Consequently, the proposed UPSM constellation can be constructed by power allocation and label-to-symbol mapping optimization. Second, this paper proposes an improved protograph LDPC code with the aid of a protograph extrinsic information transfer (PEXIT) algorithm and an asymptotic weight distribution (AWD) function. In particular, some empirical constraints (including matrix dimension and variable node degree distribution) are imposed on the protograph (i.e., the base matrix) to reduce the encoding and decoding complexity. Next, based on a computer search method, the PEXIT algorithm optimizes the protograph LDPC code to achieve the minimum decoding threshold, leading to substantial bit-error-rate (BER) performance in the low signal-to-noise ratio (SNR) region. Furthermore, to avoid the error floor in the high SNR region, the AWD function is employed during the construction of the improved protograph LDPC code, guaranteeing that the protograph enables the linear minimum distance growth property.

The simulation and analysis results show that the proposed UPSM constellation mapping scheme considerably outperforms the natural constellation, Gray-like label (GL) constellation, and unequal power GL (UPGL) constellation. In addition, the proposed improved protograph LDPC code exhibits excellent convergence performance and the lowest decoding threshold compared with the existing counterparts in the MIMO -GSM-VLC system with the proposed UPSM mapping scheme.

This paper conducts an in-depth investigation of the joint design of protograph LDPC codes and spatial MPPM constellation. The proposed constellation mapping scheme and the improved protograph LDPC code benefit from the remarkable BER performance and strong antifading robustness. Given these advantages, the proposed schemes are expected to be competitive solutions for indoor VLC applications.