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The Ising channel, which was introduced in 1990, is a channel with memorythat models Inter-Symbol interference. In this paper we consider the Isingchannel with feedback and find the capacity of the channel together with acapacity-achieving coding scheme. To calculate the channel capacity, anequivalent dynamic programming (DP) problem is formulated and solved. Using theDP solution, we establish that the feedback capacity is the expression$C=(\frac{2H_b(a)}{3+a})\approx 0.575522$ where $a$ is a particular root of afourth-degree polynomial and $H_b(x)$ denotes the binary entropy function.Simultaneously, $a=\arg \max_{0\leq x \leq 1} (\frac{2H_b(x)}{3+x})$. Finally,a simple, error-free, capacity-achieving coding scheme is provided togetherwith outlining a strong connection between the DP results and the codingscheme.

Capacity and coding for the Ising Channel with Feedback