Explicit Constructions for Type-1 QC-LDPC Codes With Girth 12

Given any <inline-formula> <tex-math notation="LaTeX">$J\times J\,\,(J>3)$ </tex-math></inline-formula> square matrix over <inline-formula> <tex-math notation="LaTeX">$\textbf {Z}_{P}$ </tex-math></inline-formula> such that the differences of any two row vectors contain each element in <inline-formula> <tex-math notation="LaTeX">$\textbf {Z}_{P}$ </tex-math></inline-formula> at most once, a class of <inline-formula> <tex-math notation="LaTeX">$(3,L)$ </tex-math></inline-formula>-regular quasi-cyclic low-density parity-check codes is explicitly constructed with lengths <inline-formula> <tex-math notation="LaTeX">$PJL^{2}$ </tex-math></inline-formula> and girth 12, where <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> is any integer satisfying <inline-formula> <tex-math notation="LaTeX">$3<L\leq J$ </tex-math></inline-formula>. Simulation results show that the new codes perform very well for moderate rates and lengths.