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

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 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 19p are determined. In order to show the good performance of this class of QC-LDPC codes, numerical results are also provided.