Notes 9 : Expander codes , beginning of list decoding

Definition 1 (distance amplified code G(C)) Let G = (L,R,E) be a bipartite graph with L = [n], R = [m], which is D-left-regular and d-right-regular. Let C be a binary linear code of block length n = |L|. For c ∈ {0, 1}n, define G(c) ∈ ({0, 1}d)m by G(c)j = (cΓ1(j), cΓ2(j), · · · , cΓd(j)), for j ∈ [m], where Γi(j) ∈ L denotes the i-th neighbor of j ∈ R. Now define the code G(C) as G(C) = {G(c)|c ∈ C}. Since each bit of a codeword c ∈ C is repeated D times in the associated codeword G(c) ∈ G(C), we have