Explicit Binary Tree Codes with Sub-logarithmic Size Alphabet
PROCEEDINGS OF THE 54TH ANNUAL ACM SIGACT SYMPOSIUM ON THEORY OF COMPUTING (STOC '22)(2022)
摘要
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.. 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( 1/log n) tree code. In this work we improve upon these prior works and construct a distance-delta tree code with (log n)(O (root 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 (1/( log log n)(2)), exponentially improving upon the above-mentioned work by Gelles et al.
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关键词
explicit constructions,tree codes
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