Explicit Binary Tree Codes with Polylogarithmic Size Alphabet

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.