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 $p\equiv 1~(\bmod ~57)$ , 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.