Relation Between GCD Constraint and Full-Length Row-Multiplier QC-LDPC Codes With Girth Eight

The greatest-common-divisor (GCD) method is a general framework employing a set of simple inequalities (called GCD constraint) to guarantee girth eight for a class of
$(J,L)$
quasi-cyclic (QC) low-density parity-check (LDPC) codes. However, an important problem, i.e., whether the GCD constraint is necessary for this class of codes to have girth eight, remains open. In this letter, the question is answered affirmatively, following which a novel algorithm aiming to find the shortest codes with girth eight in such a class is proposed. Besides, a close connection is established between the GCD method and the base expansion method, which are both applicable for any
$J$
and any
$L$
.