An explicit method for construction of CTBC codes with girth 6

A cylinder-type block-circulant (CTBC) code with circulant permutation matrix (CPM) size p is presented by a binary base matrix with a cylindrical structure and a slope-vector on Zp. The slope-vector is called proper if the corresponding CTBC code is 4-cycle free. In this paper, an algorithm is presented that for a given base matrix with a cylindrical structure, explicitly generates a proper slope-vector. This proper slope-vector provides a lower-bound on the CPM-sizes of the corresponding CTBC codes required to achieve girth at least 6. Interestingly, the lower bound, minimum distance and 6-cycle multiplicity of the constructed CTBC codes are better than some masked and unmasked known explicitly constructed 4-cycle free QC LDPC codes with the same regularity. Simulation results show that the constructed CTBC codes outperform some 4-cycle free QC LDPC codes in binary and nonbinary cases.