6.440 Essential Coding Theory Lecture 18 1 Notations and Main Results from Last Lecture

Let G = L ∪ R be a bipartite graph with n left vertices and m right vertices. Then G corresponds to a binary code CG with block length n, and dimension k ≥ n −m, where the left vertices correspond to variables x1, . . . , xn ∈ {0, 1}, while the right vertices c1, . . . , cm correspond to constraints, such that ci is satisfied iff ∑ u↔ci xu = 0. Let SAT be the set of satisfied constraints, and UNSAT be its complement in R. For u ∈ L let sat(u) be the number of satisfied constraints that u is adjacent to, and let unsat(u) be the number of unsatisfied such constraints. The graph G is (c, d)-bounded if the left degrees are at most c and the right degrees are at most d. Let Γ(S) = {v ∈ R | ∃ u ∈ S, u ↔ v}. G is a (γ, δ)expander if: ∀S ⊆ L s.t. |S| < δn, it is the case that |Γ(S)| > γ|S|. We assume γ > 1. We also needed a special subset of Γ(S) namely, the set of vertices with unique neighbors in S, denoted Γ(S) = {v ∈ R | ∃ ! u ∈ S, u ↔ v}. As before, G is a (γ̃, δ)unique neighbor expander if: ∀S ⊆ L s.t. |S| < δn, it is the case that |Γ(S)| > γ̃|S|. We showed last time the existence of such expander codes and exhibited a relation between the expansion parameters of the underlying graph G and the relative distance δ(CG) of the resulting Gallager code. We could thus obtain binary codes with rate > 0 and relative minimum distance > 0, which is a non-trivial task.