  Classically, coding theory has been concerned with the problem of
transmitting a single message in a format which is robust to noise. Recently,
researchers have turned their attention to designing coding schemes to make
two-way conversations robust to noise. That is, given an interactive
communication protocol $\Pi$, an \emph{interactive coding scheme} converts
$\Pi$ into another communication protocol $\Pi'$ such that, even if errors are
introduced during the execution of $\Pi'$, the parties are able to determine
what the outcome of running $\Pi$ would be in a noise-free setting.
  We consider the problem of designing interactive coding schemes which allow
the parties to simulate the original protocol using little memory.
Specifically, given any communication protocol $\Pi$ we construct robust
simulating protocols which tolerate a constant noise rate and require the
parties to use only $O(\log d \log s)$ memory, where $d$ is the depth of $\Pi$
and $s$ is a measure of the size of $\Pi$. Prior to this work, all known coding
schemes required the parties to use at least $\Omega(d)$ memory, as the parties
were required to remember the transcript of the conversation thus far.
Moreover, our coding scheme achieves a communication rate of
$1-O(\sqrt{\varepsilon})$ over oblivious channels and
$1-O(\sqrt{\varepsilon\log\log\tfrac{1}{\varepsilon}})$ over adaptive
adversarial channels, matching the conjecturally optimal rates. Lastly, we
point to connections between fault-tolerant circuits and coding for interactive
communication with small memory.
