Necessary and Sufficient Girth Conditions for LDPC Tanner Graphs with Denser Protographs

TL;DR

This paper establishes necessary and sufficient girth conditions for QC LDPC codes with column weight 4, enabling efficient code construction with girth between 6 and 12, based on algebraic properties of the parity-check matrix.

Contribution

It extends girth condition analysis to denser protographs with column weight 4, providing algorithms and insights for code design and girth control.

Findings

Girth conditions are linked to properties of matrix powers and circulant submatrices.

Algorithms for constructing codes with specific girths are presented.

Girth conditions can be checked via algebraic properties of the parity-check matrix.

Abstract

This paper gives necessary and sufficient conditions for the Tanner graph of a quasi-cyclic (QC) low-density parity-check (LDPC) code based on the all-one protograph to have girth 6, 8, 10, and 12, respectively, in the case of parity-check matrices with column weight 4. These results are a natural extension of the girth results of the already-studied cases of column weight 2 and 3, and it is based on the connection between the girth of a Tanner graph given by a parity-check matrix and the properties of powers of the product between the matrix and its transpose. The girth conditions can be easily incorporated into fast algorithms that construct codes of desired girth between 6 and 12; our own algorithms are presented for each girth, together with constructions obtained from them and corresponding computer simulations. More importantly, this paper emphasizes how the girth conditions of the Tanner graph corresponding to a parity-check matrix composed of circulants relate to the matrix obtained by adding (over the integers) the circulant columns of the parity-check matrix. In particular, we show that imposing girth conditions on a parity-check matrix is equivalent to imposing conditions on a square circulant submatrix of size 4 obtained from it.