Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE Access

Tanner (3, 23)-Regular QC-LDPC Codes: Cycle Structure and Girth Distribution

Qi Wang, Jingping Che, Huaan Li, Zhen Luo, Bo Zhang, Hui Liu

Published Jan 1, 2024• 6 citations
Publisher

Abstract

This paper studies on a class of quasi-cyclic LDPC (QC-LDPC) codes, i.e., Tanner (3, 23)-regular QC-LDPC codes of code length 23pwithpbeing a prime andp≡ 1 (mod 69). We first analyze the cycle structure of Tanner (3, 23)-regular QC-LDPC codes, and divide their cycles of lengths 4, 6, 8, and 10 into five equivalent types. We propose the sufficient and necessary condition for the existence of these five types of cycles, i.e., the polynomial equations in a 69th unit root of the prime field Fp. We check the existence of solutions for such polynomial equations by using the Euclidean division algorithm and obtain the candidate girth values of Tanner (3, 23)-regular QC-LDPC codes. We summarize the results and determine the girth distribution of Tanner (3, 23)-regular QC-LDPC codes.

DOI
10.1109/access.2024.3355926

Related papers

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

Hengzhou Xu, Yake Duan, Xiaoxiao Miao +1 more

In this paper, the girth distribution of the Tanner’s (3, 17)-regular quasi-cyclic LDPC (QC-LDPC) codes with code length$17p$is determined, where$p$is a prime and$p \equiv 1~(\bmod ~51)$. 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$\mathbb {F}_{p}$. 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.

IEEE Access•Jan 1, 2019•8 citations

A Note on the Girth of (3, 19)-Regular Tanner’s Quasi-Cyclic LDPC Codes

Manjie Zhou, Hai Jiang Zhu, Hengzhou Xu +2 more

In this article, we study the cycle structure of (3, 19)-regular Tanner’s quasi-cyclic (QC) LDPC codes with code length$19p$, where$p$is a prime and$p\equiv 1~(\bmod ~57)$, and transform the conditions for the existence of cycles of lengths not more than 10 into polynomial equations in a 57th root of unity of the prime field$\mathbb {F}_{p}$. By employing the Euclidean division algorithm to check whether these equations have solutions over the prime field$\mathbb {F}_{p}$, the girth values of (3, 19)-regular Tanner’s QC-LDPC codes of code length$19p$are determined. In order to show the good performance of this class of QC-LDPC codes, numerical results are also provided.

IEEE Access•Jan 1, 2021•10 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.

IEEE Access•Jan 1, 2018•40 citations

Girth Analysis of Tanner (5,11) Quasi-Cyclic LDPC Codes

Hengzhou Xu, Hai Jiang Zhu, Mengmeng Xu +2 more

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 (5,11) QC-LDPC codes. We first analyze the cycles of Tanner (5,11) QC-LDPC codes, and obtain the conditions for the existence of cycles of length less than 12 in Tanner (5,11) QC-LDPC codes of length 11p where p is a prime number and p =1 (mod 55). Notice that the condition is represented by the polynomial equations in a 55th root of unity of the prime field Fp. By checking the existence of solutions for these equations over Fp, the girths of Tanner (5,11) QC-LDPC codes are obtained.

International Conference on Computational Intelligence and Security (CIS)•Nov 1, 2018•5 citations

Girth of Tanner’s (3, 5) Quasi-Cyclic LDPC Codes

Sunghwan Kim, Jong‐Seon No, Habong Chung +1 more

In this paper, the cycles of Tanner’s (3, 5) quasi-cyclic (QC) low-density parity-check (LDPC) codes are analyzed and their girth values are derived. The conditions for the existence of cycles of lengths 4, 6, 8, and 10 in Tanner’s (3, 5) QC LDPC codes of length 5p are expressed in terms of polynomial equations in a 15-th root of unity of the prime field Fp. By checking the existence of solutions for these equations over Fp, the girths of Tanner’s (3, 5) QC LDPC codes are derived.

한국통신학회 학술대회 및 강연회•Apr 1, 2005•0 citations

On the girth of tanner (3, 5) quasi-cyclic LDPC codes

Sunghwan Kim, Jong‐Seon No, Habong Chung +1 more

In this correspondence, the cycles of Tanner (3,5) quasi-cyclic (QC) low-density parity-check (LDPC) codes are analyzed and their girth values are derived. The conditions for the existence of cycles of lengths 4,6,8, and 10 in Tanner (3,5) QC LDPC codes of length 5p are expressed in terms of polynomial equations in a 15th root of unity of the prime field F/sub p/. By checking the existence of solutions for these equations over F/sub p/, the girths of Tanner (3,5) QC LDPC codes are derived.

IEEE Transactions on Information Theory•Apr 1, 2006•33 citations
This paper studies on a class of quasi-cyclic LDPC (QC-LDPC) codes, i.e., Tanner (3, 23)-regular QC-LDPC codes of code length 23pwithpbeing a prime andp≡ 1 (mod 69). We first analyze the cycle structure of Tanner (3, 23)-regular QC-LDPC codes, and divide their cycles of lengths 4, 6, 8, and 10 into five equivalent types. We propose the sufficient and necessary condition for the existence of these five types of cycles, i.e., the polynomial equations in a 69th unit root of the prime field Fp. We check the existence of solutions for such polynomial equations by using the Euclidean division algorithm and obtain the candidate girth values of Tanner (3, 23)-regular QC-LDPC codes. We summarize the results and determine the girth distribution of Tanner (3, 23)-regular QC-LDPC codes.
In this paper, the girth distribution of the Tanner’s (3, 17)-regular quasi-cyclic LDPC (QC-LDPC) codes with code length17p17p17pis determined, wherepppis a prime andp≡1 ( mod  51)p \equiv 1~(\bmod ~51)p≡1 (mod 51). 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 fieldFp\mathbb {F}_{p}Fp​. 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.
In this article, we study the cycle structure of (3, 19)-regular Tanner’s quasi-cyclic (QC) LDPC codes with code length19p19p19p, wherepppis a prime andp≡1 ( mod  57)p\equiv 1~(\bmod ~57)p≡1 (mod 57), and transform the conditions for the existence of cycles of lengths not more than 10 into polynomial equations in a 57th root of unity of the prime fieldFp\mathbb {F}_{p}Fp​. By employing the Euclidean division algorithm to check whether these equations have solutions over the prime fieldFp\mathbb {F}_{p}Fp​, the girth values of (3, 19)-regular Tanner’s QC-LDPC codes of code length19p19p19pare determined. In order to show the good performance of this class of QC-LDPC codes, numerical results are also provided.
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 length13p13p13pforpppbeing a prime of the form(39m+1)(39m+1)(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 filedFp\mathbb {F}_{p}Fp​. By solving these polynomial equations overFp\mathbb {F}_{p}Fp​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.
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 (5,11) QC-LDPC codes. We first analyze the cycles of Tanner (5,11) QC-LDPC codes, and obtain the conditions for the existence of cycles of length less than 12 in Tanner (5,11) QC-LDPC codes of length 11p where p is a prime number and p =1 (mod 55). Notice that the condition is represented by the polynomial equations in a 55th root of unity of the prime field Fp. By checking the existence of solutions for these equations over Fp, the girths of Tanner (5,11) QC-LDPC codes are obtained.