Explicit Constructions of Girth-Eight QC-LDPC Codes: A Unified Framework Motivated by TDGS

A new search method named two-dimensional greedy search (TDGS) is proposed to progressively determine each element within an exponent matrix, so as to generate quasi-cyclic (QC) low-density parity-check (LDPC) codes without 4-cycles and 6-cycles. By applying four different two-dimensional scanning orders to the TDGS, a design framework is formulated to unify several existing and novel explicit constructions for short (3, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L$</tex>)-regular QC-LDPC codes without 4-cycles and 6-cycles. Through equivalence analysis between the exponent matrices greedily found by the TDGS and directly defined by explicit constructions, novel girth properties of (and connections between) these existing explicit constructions are also revealed.