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

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.