Tanner (3, 23)-Regular QC-LDPC Codes: Cycle Structure and Girth Distribution

This paper studies a class of quasi-cyclic LDPC (QC-LDPC) codes, i.e., Tanner (3, 23)-regular QC-LDPC codes of code length
$23p$
with
$p$
being a prime and
$p \equiv 1 (\mathrm {mod} 69)$
. 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
$\mathbb {F}_{p}$
. 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.