Detecting cycles of length $10$ in the Tanner graph of a QC-LDPC code based on protograph analysis

We complete the description of the procedure of topological expansion of a bipartite graph without parallel branches on the plane of changing the structure of cycles of length up to $10$ inclusive. Based on previous papers, we have extended a set of theorems specifying transformation rules for cycles and paths when passing from a protograph to the Tanner graph. We propose a procedure for detecting the existence of a cycle of length $10$ in the expanded graph by analyzing the protograph.