The Maximum-Likelihood Decoding Threshold for Cycle Codes of Graphs

Abstract

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