Constructing good QC-LDPC codes by pre-lifting protographs

Quasi-cyclic (QC) low-density parity-check (LDPC) codes are of great interest to code designers because of their implementation advantages and algebraic properties that facilitate their analysis. In this paper, we present some new results on QC-LDPC codes that are constructed using a two-step lifting procedure based on a protograph, and, by implementing this method instead of the usual one-step procedure, we are able to show improved minimum distance and girth properties. We also present two design rules to construct QC-LDPC codes: one uses only circulant permutation matrices at the first (pre-lifting) stage and the other uses a selection of non-commuting permutation matrices. For both techniques, we obtain a demonstrable increase in the minimum distance compared to a one-step circulant-based lifting. The expected performance improvement is verified by simulation results.