Quasi-cyclic LDPC (QC-LDPC) codes have been accepted as the standard codes of 5G enhanced mobile broadband data channel. These standard codes are designed to support multiple lifting sizes and possess rate-compatible property, which can help adapt various information lengths and code rates well. In this paper, we propose an algebra-assisted method for constructing QC-LDPC codes with such properties. We will first review the encoding mechanism and requirements of 5G LDPC codes, and present cycle analysis for such emerging codes. We then propose a metric, referred to as weighted average number of cycles (WANC), from the perspective of cycle structure for constructing the QC-LDPC codes that can support multiple lifting sizes. Based on the WANC metric and algebraic methods, we develop a simple and practical algorithm to construct this kind of QC-LDPC codes. We finally apply the proposed algorithm to construct the exponent matrices for cases of 5G LDPC codes and the standard LDPC codes of consultative committee for space data systems, respectively. Simulation results show that the proposed WANC metric and designed algorithm are feasible and effective, and thus can be utilized to design other similar QC-LDPC codes.

In this letter, we study the construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes by array dispersion and masking. We first propose a class of arrays of circulant permutation matrices (CPMs) of arbitrary size obtained by dispersion. Based on this class of arrays of CPMs, a class of QC-LDPC codes is constructed. For these resulting codes, we analyze their cycle structure and derive the numbers, which indicate how many cycles pass through each CPM in their parity-check matrices. Following the indication numbers with the order from largest to smallest, we replace the corresponding CPMs with zero matrices (ZMs) of the same size one by one until a targeted masking termination condition is satisfied. Then, the resultant arrays of CPMs and ZMs give a large class of QC-LDPC codes. Numerical results show that the codes constructed perform well over the additive white Gaussian noise (AWGN) channel when decoded with the sum-product algorithm.

This paper presents a nonbinary low-density parity-check (LDPC) coded spatial modulation (CSM) for multiple-input multiple-output communication systems, in which the information bits for choosing active transmit antennas and the bits for choosing constellation signals are protected by a nonbinary LDPC code. We apply a many-to-one mapping known as Gallager mapping to signal constellation, resulting in an improved constellation design for the CSM system. Furthermore, we propose a Gallager mapping-based scheme to solve the design problem of the CSM system with arbitrary transmit antennas, in which Gallager mapping is used to map the index information bits to the active antenna, thus an integer-bit transmission can be achieved for any number of transmit antennas. With the use of nonbinary LDPC codes, a non-iterative receiver is able to recover the undistinguished information caused by the many-to-one mapping. Several communication scenarios are studied and simulation results show that the proposed scheme can offer substantial performance gains with design flexibility over the Rayleigh fading channel.

In this letter, we consider the construction of q-ary low-density parity-check (LDPC) codes by jointly optimizing the girth and number of shortest cycles in the frame of superposition. We begin with constructing binary quasi-cyclic LDPC codes by superposition. By replacing the nonzero entries of the parity-check matrices of the resulting binary codes with the nonzero elements of GF(q), a class of q-ary LDPC codes is obtained. Furthermore, we analyze the cycle structure of the constructed codes for a given degree distribution and code length, and present a search algorithm for finding q-ary LDPC codes with few cycles of length g, where g is the optimized girth value. Simulation results show that the proposed codes perform well under the iterative decoding algorithms.

Girth plays an important role in the design of low-density parity-check (LDPC) codes. Motivated by the works on the girth of some classes of Tanner quasi-cyclic (QC) LDPC codes, e.g., Tanner (3,5), (3,7), (3,11), and (5,7) codes, we in this paper study the girth of Tanner (J,L) QC-LDPC codes where J and L can be any two positive integers. According to the sufficient and necessary conditions for the existence of cycles of lengths 4, 6, 8, and 10, we propose an algorithm to determine the girth of Tanner (J,L) QC-LDPC codes with finite code lengths. Through the analysis of the obtained girth values, we generalize the laws of the girth distributions of Tanner (J,L) QC-LDPC codes. Furthermore, based on the exponent matrices of Tanner (J,L) QC-LDPC codes with known girths, we employ the column selection method and/or the masking technique to construct binary/nonbinary LDPC codes. Numerical results show that the constructed LDPC codes have good performance under iterative decoding over the additive white Gaussian noise (AWGN) channel.

Quasi‐cyclic (QC) LDPC codes play an important role in 5G communications and have been chosen as the standard codes for 5Genhanced mobile broadband(eMBB) data channel. In this paper, we study the construction of QC LDPC codes based on an arbitrary given expansion factor (or lifting degree). First, we analyze the cycle structure of QC LDPC codes and give the necessary and sufficient condition for the existence of short cycles. Based on the fundamental theorem of arithmetic in number theory, we divide the integer factorization into three cases and present three classes of QC LDPC codes accordingly. Furthermore, a general construction method of QC LDPC codes with girth of at least 6 is proposed. Numerical results show that the constructed QC LDPC codes perform well over the AWGN channel when decoded with the iterative algorithms.