On Counting Short Cycles of LDPC Codes Using the Tanner Graph Spectrum

Abstract

Counting short cycles in bipartite graphs is a fundamental problem of interest in the analysis and design of low-density parity-check (LDPC) codes. The vast majority of research in this area is focused on algorithmic techniques. Most recently, Blake and Lin proposed a computational technique to count the number of cycles of length g in a bi-regular bipartite graph, where g is the girth of the graph. The information required for the computation is the node degree and the multiplicity of the nodes on both sides of the partition, as well as the eigenvalues of the adjacency matrix of the graph (graph spectrum). In this work, we extend this result. First, we derive a similar result to compute the number of cycles of length g + 2, …, 2g − 2, for bi-regular bipartite graphs. Second, using a counter-example, we demonstrate that the information of the degree distribution and the spectrum of a bi-regular bipartite graph is, in general, insufficient to count the cycles of length i ≥ 2g.