  This paper proposes a new method of constructing compact fully-connected
Quasi-Cyclic Low Density Parity Check (QC-LDPC) codes with girth g = 8, 10, and
12. The originality of the proposed method is to impose constraints on the
exponent matrix P to reduce the search space drastically. For a targeted
lifting degree of N, the first step of the method is to sieve the integer ring
Z_N to make a particular sub-group with specific properties to construct the
second column of P (the first column being filled with zeros). The remaining
columns of P are determined recursively as multiples of the second column by
adapting the sequentially multiplied column (SMC) method whereby a controlled
greedy search is applied at each step. The codes constructed with the proposed
semi-algebraic method show lengths that can be significantly shorter than their
best counterparts in the literature.
