Construction of girth-8 (3,L)-QC-LDPC codes of smallest CPM size using column multipliers

In this paper, a new method for the construction of the exponent matrix of quasi-cyclic low-density parity-check (QC-LDPC) codes is proposed. The entries of the exponent matrix are based on the column multipliers. To find the column multipliers, a parameter Sα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varvec{S}_{\varvec{\alpha}}} $$\end{document} is defined which gives the value of column multiplier of the α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varvec{\alpha}} $$\end{document}th column. The proposed method reduced the complexity related to the formation of the exponent matrix and results in (3,L)-QC-LDPC codes with girth at least eight, for L>3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varvec{L} > 3} $$\end{document}. Also, a lower bound on the size of the circulant permutation matrix (CPM) for a QC-LDPC code is derived, and the codes constructed by this method are optimal to the given bound. Further, most of the codes constructed using this method are of smaller CPM size. Specifically, for L>25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varvec{L} > 25} $$\end{document}, our constructed QC-LDPC codes have the shortest CPM size compared to the existing ones in the literature.