Towards deterministic tree code constructions

We present a deterministic operator on tree codes -- we call tree code product -- that allows one to deterministically combine two tree codes into a larger tree code. Moreover, if the original tree codes are efficiently encodable and decodable, then so is their product. This allows us to give the first deterministic subexponential-time construction of explicit tree codes: we are able to construct a tree code <i>T</i> of size <i>n</i> in time 2^{<i>n</i>^ε,}. Moreover, <i>T</i> is also encodable and decodable in time 2^{<i>n</i>^ε,}. We then apply our new construction to obtain a deterministic constant-rate error-correcting scheme for interactive computation over a noisy channel with random errors. If the length of the interactive computation is <i>n</i>, the amount of computation required is deterministically bounded by <i>n</i>^{1+<i>o</i>(1)}, and the probability of failure is <i>n</i>^-ω(1).