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.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.

IEEE AccessDec 10, 201821 citations

On the Girth of Tanner (3, 13) Quasi-Cyclic LDPC Codes

Hengzhou Xu, Huaan Li, Dan Feng +2 more

Girth is an important structural property of low-density parity-check (LDPC) codes. Motivated by the works on the girth of Tanner (3, 5), (3, 7), (3, 11), and (5, 7) quasi-cyclic (QC) LDPC codes, we, in this paper, study the girth of Tanner (3, 13) QC-LDPC codes of length $13p$ for $p$ being a prime of the form $(39m+1)$ . First, the cycle structure of Tanner (3, 13) QC-LDPC codes is analyzed, and the cycles of length lesser than 12 are divided into five equivalent classes. Based on each equivalent class, the existence of the cycles is equivalent to the solution of polynomial equations in a 39th unit root in the prime filed $\mathbb {F}_{p}$ . By solving these polynomial equations over $\mathbb {F}_{p}$ and summarizing the resulting candidate prime values, the girth of Tanner (3, 13) QC-LDPC codes is obtained. As an advantage, Tanner (3, 13) QC-LDPC codes have much higher code rates than Tanner (3, 5), (3, 7), (3, 11), and (5, 7) QC-LDPC codes, and provide a promising coding scheme for the data storage systems and optical communications.Girth is an important structural property of low-density parity-check (LDPC) codes. Motivated by the works on the girth of Tanner (3, 5), (3, 7), (3, 11), and (5, 7) quasi-cyclic (QC) LDPC codes, we, in this paper, study the girth of Tanner (3, 13) QC-LDPC codes of length$13p$for$p$being a prime of the form$(39m+1)$. First, the cycle structure of Tanner (3, 13) QC-LDPC codes is analyzed, and the cycles of length lesser than 12 are divided into five equivalent classes. Based on each equivalent class, the existence of the cycles is equivalent to the solution of polynomial equations in a 39th unit root in the prime filed$\mathbb {F}_{p}$. By solving these polynomial equations over$\mathbb {F}_{p}$and summarizing the resulting candidate prime values, the girth of Tanner (3, 13) QC-LDPC codes is obtained. As an advantage, Tanner (3, 13) QC-LDPC codes have much higher code rates than Tanner (3, 5), (3, 7), (3, 11), and (5, 7) QC-LDPC codes, and provide a promising coding scheme for the data storage systems and optical communications.

IEEE AccessJan 1, 201840 citations

Construction of good quasi-cyclic LDPC codes

Xiao Xiao, Lee

Jan 1, 200616 citations

Calculation of Girth of Tanner Graph in LDPC Codes

M. Ahmadi, H. Dehghani, ‎Saeid Alikhani +1 more

Trends in Applied Sciences ResearchNov 1, 20124 citations

Tanner

(J,L)

Quasi-Cyclic LDPC Codes: Girth Analysis and Derived Codes

(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.

13p

for

p

being a prime of the form

(39m+1)

. First, the cycle structure of Tanner (3, 13) QC-LDPC codes is analyzed, and the cycles of length lesser than 12 are divided into five equivalent classes. Based on each equivalent class, the existence of the cycles is equivalent to the solution of polynomial equations in a 39th unit root in the prime filed

\mathbb {F}_{p}

. By solving these polynomial equations over

\mathbb {F}_{p}

and summarizing the resulting candidate prime values, the girth of Tanner (3, 13) QC-LDPC codes is obtained. As an advantage, Tanner (3, 13) QC-LDPC codes have much higher code rates than Tanner (3, 5), (3, 7), (3, 11), and (5, 7) QC-LDPC codes, and provide a promising coding scheme for the data storage systems and optical communications.