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For a class $\mathcal{C}$ of binary linear codes, we write$\theta_{\mathcal{C}}\colon (0,1) \to [0,\frac{1}{2}]$ for themaximum-likelihood decoding threshold function of $\mathcal{C}$, the functionwhose value at $R \in (0,1)$ is the largest bit-error rate $p$ that codes in$\mathcal{C}$ can tolerate with a negligible probability of maximum-likelihooddecoding error across a binary symmetric channel. We show that, if$\mathcal{C}$ is the class of cycle codes of graphs, then$\theta_{\mathcal{C}}(R) \le \frac{(1-\sqrt{R})^2}{2(1+R)}$ for each $R$, andshow that equality holds only when $R$ is asymptotically achieved by cyclecodes of regular graphs.

The maximum-likelihood decoding threshold for graphic codes