Seminar on Error Correcting Codes: Unbalanced Expanders from Parvaresh-Vardy Codes

Expanders are graphs that are sparse yet highly connected. The connectivity of the expander is measured by its vertex expansion, which asks that every set S of vertices that is not too large has significantly more than |S| neighbors. Definition. A bipartite (multi)graph with N left-vertices, M right-vertices, and left degree D is specified by a function Γ : [N ]× [D]→ [M ], where Γ(x, y) denotes the y-th neighbor of x. For a set S ⊆ [N ], we write Γ(S) to denote its set of neighbors {Γ(x, y) : x ∈ S, y ∈ [D]}.