Tanner $(J,L)$ Quasi-Cyclic LDPC Codes: Girth Analysis and Derived Codes

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. The numerical results show that the constructed LDPC codes have good performance under iterative decoding over the additive white Gaussian noise channel.