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

For every constant 1 we give an explicit binary tree code with distance 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.