Inbar Ben Yaacov
ABSTRACT
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 (logn)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/logn) tree code.
In this work we improve upon these prior works and construct a distance-δ tree code with (logn)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/(loglogn)2), exponentially improving upon the above-mentioned work by Gelles et al.
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Index Terms
- Explicit binary tree codes with sub-logarithmic size alphabet
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