Explicit binary tree codes with sub-logarithmic size alphabet

Since they were first introduced by Schulman (STOC 1993), the construction of tree codes remained an elusive open problem. The state-of-the-art construction by Cohen, Haeupler and Schulman (STOC 2018) has constant distance and (log n) e

colors for some constant e > 1 that depends on the distance, where n is the depth of the tree. Insisting on a constant number of colors at the expense of having vanishing distance, Gelles, Haeupler, Kol, Ron-Zewi, and Wigderson (SODA 2016) constructed a distance Ω( 1 log n ) tree code. In this work we improve upon these prior works and construct a distance-δ tree code with (log n) O( √ δ) colors. This is the first construction of a constant distance tree code with sub-logarithmic number of colors. Moreover, as a direct corollary we obtain a tree code with a constant number of colors and distance Ω 1 (log log n) 2 ! , exponentially improving upon the above-mentioned work by Gelles et al.