This paper proposes a new method of construction of compact fully-connected Quasi-Cyclic Low Density Parity Check (QC-LDPC) code with girth g = 10 and g = 12. The originality of the proposed method is to impose constraint on the exponent matrix P to reduce the search space drastically. For a targeted expansion factor of N, the first step of the method is to sieve the integer ring Z_N to make a particular subgroup 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 thanks to an adaptation of the sequentially multiplied column (SMC) method where a controlled greedy search is applied at each step. The codes constructed with the proposed semi-algebraic method have lengths that can be significantly shorter than the best counterparts in the literature. To illustrate the great potential of the SMC method, we give the explicit construction of a rate 0.75 irregular LDPC code of size 65, 220 that allows a gain of 0.15 dB compared to the code of same rate and size 64,800 of the DVB-S2.