This paper makes progress on the problem of explicitly constructing a binary tree code with constant distance and constant alphabet size.

For every constant $\delta < 1$ we give an explicit binary tree code with distance $\delta$ and alphabet size $poly(\log{n})$, where $n$ is the depth of the tree. This is the first improvement over a two-decade-old construction that has an exponentially larger alphabet of size $poly(n)$.

As part of the analysis, we prove a bound on the number of positive integer roots a real polynomial can have in terms of its sparsity with respect to the Newton basis - a result of independent interest.

Significant changes in presentation; Construction with distance larger than one half.

This paper makes progress on the problem of explicitly constructing a binary tree code with constant distance and constant alphabet size.

For every constant $\delta < 1$ we give an explicit binary tree code with distance $\delta$ and alphabet size $(\log{n})^{O(1)}$, where $n$ is the depth of the tree. This is the first improvement over a two-decade-old construction that has an exponentially larger alphabet of size $n^{O(1)}$.

As part of the analysis, we prove a bound on the number of positive integer roots a real polynomial can have in terms of its sparsity with respect to the Newton basis - a result of independent interest.