Necessary and Sufficient Girth Conditions for Tanner Graphs of Quasi-Cyclic LDPC Codes

TL;DR

This paper establishes necessary and sufficient girth conditions for protograph-based LDPC codes, linking girth properties to matrix substructures, and provides algorithms for constructing codes with specific girths, supported by simulations.

Contribution

It introduces a novel girth characterization via matrix submatrices and offers practical algorithms for designing codes with desired girth properties.

Findings

Girth conditions are equivalent to constraints on square submatrices.

Cycle structures in Tanner graphs correspond to cycles in submatrices.

Constructed codes demonstrate the effectiveness of the girth conditions.

Abstract

This paper revisits the connection between the girth of a protograph-based LDPC code given by a parity-check matrix and the properties of powers of the product between the matrix and its transpose in order to obtain the necessary and sufficient conditions for a code to have given girth between 6 and 12, and to show how these conditions can be incorporated into simple algorithms to construct codes of that girth. To this end, we highlight the role that certain submatrices that appear in these products have in the construction of codes of desired girth. In particular, we show that imposing girth conditions on a parity-check matrix is equivalent to imposing conditions on a square submatrix obtained from it and we show how this equivalence is particularly strong for a protograph based parity-check matrix of variable node degree 2, where the cycles in its Tanner graph correspond one-to-one to the cycles in the Tanner graph of a square submatrix obtained by adding the permutation matrices (or products of these) in the composition of the parity-check matrix. We end the paper with exemplary constructions of codes with various girths and computer simulations. Although, we mostly assume the case of fully connected protographs of variable node degree 2 and 3, the results can be used for any parity-check matrix/protograph-based Tanner graph.