<p style='text-indent:20px;'>The zero-error capacity of a noisy channel is defined as the least upper bound of rate at which it is possible to transmit information with zero probability of error. It was posed by Shannon and extended to channels with memories by Ahlswede, Cai and Zhang. In this paper, we give a first step towards the zero-error capacity problems of binary channels with 2-memories, and determine the zero-error capacity of at least <inline-formula><tex-math

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... $ 2^{24} $\end{document}</tex-math></inline-formula> possible cases in all <inline-formula><tex-math id="M2">\begin{document}$ 2^{28} $\end{document}</tex-math></inline-formula> cases.</p>