2021/05/07 by Roxana Smarandache, David G. M. Mitchell, Smarandache, Roxana +1
Computer Science · Engineering · #Advanced Wireless Communication Techniques #Cooperative Communication and Network Coding #Error Correcting Code Techniques #FOS: Computer and information sciences #Information Theory (cs.IT)
paper · pdf · doi:10.48550/arxiv.2105.03462
openalex publication_date 2021/05/07 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
This paper revisits the connection between the girth of a protograph-based\nLDPC code given by a parity-check matrix and the properties of powers of the\nproduct between the matrix and its transpose in order to obtain the necessary\nand sufficient conditions for a code to have given girth between 6 and 12, and\nto show how these conditions can be incorporated into simple algorithms to\nconstruct codes of that girth. To this end, we highlight the role that certain\nsubmatrices that appear in these products have in the construction of codes of\ndesired girth. In particular, we show that imposing girth conditions on a\nparity-check matrix is equivalent to imposing conditions on a square submatrix\nobtained from it and we show how this equivalence is particularly strong for a\nprotograph based parity-check matrix of variable node degree 2, where the\ncycles in its Tanner graph correspond one-to-one to the cycles in the Tanner\ngraph of a square submatrix obtained by adding the permutation matrices (or\nproducts of these) in the composition of the parity-check matrix. We end the\npaper with exemplary constructions of codes with various girths and computer\nsimulations. Although, we mostly assume the case of fully connected protographs\nof variable node degree 2 and 3, the results can be used for any parity-check\nmatrix/protograph-based Tanner graph.\n