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### Paper:

TR21-154 | 10th November 2021 11:07

#### Explicit Binary Tree Codes with Sub-Logarithmic Size Alphabet

TR21-154
Authors: Inbar Ben Yaacov, Gil Cohen, Tal Yankovitz
Publication: 10th November 2021 13:16
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 $\Omega(\frac1{\log n})$ tree code.
In this work we improve upon these prior works and construct a distance-$\delta$ tree code with $(\log{n})^{O(\sqrt{\delta})}$ 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 $\Omega\left(\frac1{(\log\log{n})^{2}}\right)$, exponentially improving upon the above-mentioned work by Gelles et al.