The Rate-Immediacy Barrier in Explicit Tree Code Constructions
Gil Cohen, Leonard J. Schulman, Piyush Srivastava
TL;DR
The paper formalizes a rate-immediacy barrier for explicit tree-code constructions by introducing immediacy codes, laminar partitions, and a data-processing framework. It proves a general rate upper bound that ties rate to immediacy and shows that existing constructions (e.g., CHS, EKS, GHKRZW) essentially attain the best possible rate within their immediacy classes, explaining why asymptotically good explicit tree codes remain elusive. The main finding is that constant-distance tree codes with non-trivial immediacy must have vanishing rate, and that overcoming this barrier will require fundamentally new ideas beyond recursive tilings of block codes. The results thus clarify a fundamental barrier and guide future work toward novel, non-recursive approaches to explicit asymptotically-good tree codes.
Abstract
Since the introduction of tree codes by Schulman (STOC 1993), explicit construction of such codes has remained a notorious challenge. While the construction of asymptotically-good explicit tree codes continues to be elusive, a work by Cohen, Haeupler and Schulman (STOC 2018), as well as the state-of-the-art construction by Ben Yaacov, Cohen, and Yankovitz (STOC 2022) have achieved codes with rate $Ω(1/\log\log n)$, exponentially improving upon the original construction of Evans, Klugerman and Schulman from 1994. All of these constructions rely, at least in part, on increasingly sophisticated methods of combining (block) error-correcting codes. In this work, we identify a fundamental barrier to constructing tree codes using current techniques. We introduce a key property, which we call immediacy, that, while not required by the original definition of tree codes, is shared by all known constructions and inherently arises from recursive combinations of error-correcting codes. Our main technical contribution is the proof of a rate-immediacy tradeoff, which, in particular, implies that any tree code with constant distance and non-trivial immediacy must necessarily have vanishing rate. By applying our rate-immediacy tradeoff to existing constructions, we establish that their known rate analyses are essentially optimal. More broadly, our work highlights the need for fundamentally new ideas--beyond the recursive use of error-correcting codes--to achieve substantial progress in explicitly constructing asymptotically-good tree codes.