A transform approach for constructing quasi-cyclic Euclidean geometry LDPC codes

A method for constructing quasi-cyclic Euclidean geometry (QC-EG) LDPC codes in the Fourier transform domain is presented. Given a Euclidean geometry over a finite field of characteristic 2, base matrices in the Fourier transform domain are first constructed. Then the inverse Fourier transforms of these base matrices, combined with row and column permutations, result in low-density arrays of circulant permutation matrices and/or zero matrices. The null spaces of these low-density arrays give a family of QC-EG-LDPC codes. Codes in a special subclass have large minimum distances and their Tanner graphs contain no harmful trapping sets with sizes smaller than their minimum distances.