Explicit Binary Tree Codes with Polylogarithmic Size Alphabet

In this talk, we consider the problem of explicitly constructing a binary tree code with constant distance and constant alphabet size. We present an explicit binary tree code with constant distance and alphabet size polylog(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). For analyzing our construction, we prove a bound on the number of integral roots a real polynomial can have in terms of its sparsity with respect to the Newton basis - a result of independent interest.

Joint work with Bernhard Haeupler and Leonard Schulman.