Optimal Search for Girth-8 Quasi Cyclic and Spatially Coupled Multiple-Edge LDPC Codes

Abstract

Two essential categories of LDPC codes that are more preferable to other types are quasi-cyclic LDPC (QC-LDPC) codes and spatially coupled LDPC convolutional codes (SC-LDPC-CCs) because of their excellent performance curves in waterfall and error floor regions. An efficient approach to construct these codes is protograph-based method that is categorized into two classes: single-edge (SE) and multiple-edge (ME) protographs. We, for the first time, provide a necessary and sufficient condition for exponent matrices of these codes with girth-8 and based on the ME-protographs. As a result, a lower bound on the lifting degree of girth-8 ME-QC-LDPC codes and a lower bound on the syndrome former memory order of girth-8 ME-SC-LDPC-CCs are obtained, which are tighter than the existing bounds in the literature.