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On the Girth of Tanner QC-LDPC Cycle Codes: An Algebraic Number Theory Approach

Haipeng Yu, Manjie Zhou, Zhongyang Yu, Mengmeng Xu, Hengzhou Xu

Published May 21, 2026• 0 citations

Abstract

Tanner quasi-cyclic low-density parity-check (QC-LDPC) codes form an important family of structured LDPC codes with favorable girth properties. This paper studies the girth of Tanner (2, L)-regular QC-LDPC codes (referred to as Tanner QC-LDPC cycle codes) for arbitrary integers L>2 and develops a novel algebraic number theoretic method to determine the girth for all sufficiently large primes p with p≡1(mod2L). We first analyze the case L=3 and prove that the girth is 12 for every prime p≡1(mod6) through exhaustive resultant computations. We then extend the method to arbitrary L and obtain a clear classification: when L is even, the girth is exactly 8 for all admissible primes; when L is odd, the girth attains the maximum value 12 for all sufficiently large admissible primes. The proof transforms cycle existence conditions into polynomial equations and applies resultant theory. This approach converts the infinite task of checking all primes into a finite set of algebraic checks. Numerical simulations show that the Tanner (2, 5)-regular non-binary code over GF(64) achieves a coding gain of approximately 0.2 dB over the 5G LDPC code of equivalent binary length.

DOI: 10.3390/axioms15050384

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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

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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.

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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.

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.