The maximum-likelihood decoding threshold for graphic codes

For a class C of binary linear codes, we write θC : (0, 1) → [0, 12 ] for the maximum-likelihood decoding threshold function of C, the function whose value at R ∈ (0, 1) is the largest bit-error rate p that codes in C can tolerate with a negligible probability of maximum-likelihood decoding error across a binary symmetric channel. We show that, if C is the class of cycle codes of graphs, then θC(R) ≤ (1− √ R) 2(1+R) for each R, and show that equality holds only when R is asymptotically achieved by the cycle codes of regular graphs.