As a special case of a quasi-cyclic (QC) low-density parity-check (LDPC) code, a full-length row-multiplier (FLRM) QC-LDPC code is described by a compact exponent matrix based on two sequences of integers. The codes designed by a framework known as greatest-common-divisor (GCD) method, belong to a salient class of FLRM QC-LDPC codes, which can eliminate cycles of length up to six by carefully selecting a special sequence subject to a set of simple inequalities. However, the GCD method ensures the absence of these cycles only if circulant sizes are larger than a certain threshold. By combining the existing GCD method, novel sequences and a new analysis method (based on new lemmas of circulants and integers) for modulo equations, a group of novel FLRM QC-LDPC codes free of 4-cycles and 6-cycles are explicitly proposed for column weights from three to five in this paper, which possess circulant sizes much smaller than the forgoing threshold. Simulations show that the new FLRM QC-LDPC codes with shorter lengths perform almost the same as the existing FLRM QC-LDPC codes with longer lengths, and that the novel FLRM QC-LDPC codes noticeably outperform their counterparts with (nearly) identical lengths.
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