Finding good tree code constructions is among the most natural questions in error-correcting codes over the Hamming channel. Unlike the usual setting, where one encodes a single message against adversarial errors, tree codes allow a sender to encode and send a stream, in an online manner, while protecting it from adversarial errors. Moreover, Schulman (STOC 93) showed that tree codes yield schemes which protect any two-party communication protocol from adversarial errors. Schulman asked whether there are explicit tree code families with constant distance and constant rate. While this question remains open, in a breakthrough, Cohen, Haeupler and Schulman (STOC 2018) constructed explicit constant-distance tree codes with rate $1/O(\log\log n)$, where $n$ denotes the message length, improving upon the $1/O(\log n)$ previously known. Subsequently, Ben-Yaacov, Cohen and Yankovitz (STOC 22) improved the rate-distance tradeoff, constructing explicit distance-$\delta$ tree codes with rate $1/O(\sqrt{\delta}\log\log n)$.

In this work we construct explicit distance-$\delta$ tree codes with
rate $1/O(\delta\log\log n)$. In particular, this implies explicit
tree codes with constant rate and distance $1/O(\log\log n)$, improving
upon the previously known $1/O(\log^2\log n)$. To achieve this improved
rate-distance tradeoff, new ideas are required. Specifically, we devise a construction whose key idea relies on MDS tree codes. MDS tree
codes were first considered by Pudlák (Linear Algebra Appl. 16), who gave an $\mathbb F_2$-linear MDS tree code construction with alphabet size
$2^{O(n^3)}$. As another contribution, we present such a construction
with alphabet size $2^{O(n^2)}$.