Girth Analysis of Tanner's (3, 17)-Regular QC-LDPC Codes Based on Euclidean Division Algorithm

• Published in: IEEE access Vol. 7; pp. 94917 - 94930
• Main Authors: Xu, Hengzhou, Duan, Yake, Miao, Xiaoxiao, Zhu, Hai
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Codes; Encoding; Euclidean division algorithm; girth; Hardware; Indexes; LDPC codes; Low density parity check codes; Mathematical analysis; Parity check codes; Polynomials; prime field; quasi-cyclic (QC); Technological innovation; Training
• ISSN: 2169-3536; 2169-3536
• DOI: 10.1109/ACCESS.2019.2929587

In this paper, the girth distribution of the Tanner's (3, 17)-regular quasi-cyclic LDPC (QC-LDPC) codes with code length <inline-formula> <tex-math notation="LaTeX">17p </tex-math></inline-formula> is determined, where <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula> is a prime and <inline-formula> <tex-math notation="LaTeX">p \equiv 1~(\bmod ~51) </tex-math></inline-formula>. By analyzing their cycle structure, five equivalent types of cycles with length not more than 10 are obtained. The existence of these five types of cycles is transmitted into polynomial equations in a 51st unit root of the prime field <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{p} </tex-math></inline-formula>. By using the Euclidean division algorithm to check the existence of solutions for such polynomial equations, the girth values of the Tanner's (3, 17)-regular QC-LDPC codes are obtained.