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This paper proposes a new method of constructing compact fully-connectedQuasi-Cyclic Low Density Parity Check (QC-LDPC) codes with girth g = 8, 10, and12. The originality of the proposed method is to impose constraints on theexponent matrix P to reduce the search space drastically. For a targetedlifting degree of N, the first step of the method is to sieve the integer ringZ_N to make a particular sub-group with specific properties to construct thesecond column of P (the first column being filled with zeros). The remainingcolumns of P are determined recursively as multiples of the second column byadapting the sequentially multiplied column (SMC) method whereby a controlledgreedy search is applied at each step. The codes constructed with the proposedsemi-algebraic method show lengths that can be significantly shorter than theirbest counterparts in the literature.

Integer Ring Sieve for Constructing Compact QC-LDPC Codes with Girths 8, 10, and 12