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

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
$\mathbb {Z}_{N}$
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.