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.
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.
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
13pfor
pbeing 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
Fp. By solving these polynomial equations over
Fpand 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.
In this article, we study the cycle structure of (3, 19)-regular Tanner’s quasi-cyclic (QC) LDPC codes with code length
19p, where
pis a prime and
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 field
Fp. By employing the Euclidean division algorithm to check whether these equations have solutions over the prime field
Fp, the girth values of (3, 19)-regular Tanner’s QC-LDPC codes of code length
19pare determined. In order to show the good performance of this class of QC-LDPC codes, numerical results are also provided.
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.