Self-Correction of Transmission on Regular Trees

We consider noisy binary channels on regular trees and introduce periodic enhancements consisting of locally self-correcting the signal in blocks without break of the symmetry of the model. We focus on the realistic class of within-descent self-correction realized by identifying all descendants $k$ generations down a vertex with their majority. We show that this also allows reconstruction strictly beyond the critical distortion. We further identify the limit at which the critical distortions of within-descent $k$ self-corrected transmission converge, which turns out to be the critical point for ferromagnetic Ising model on that tree. We finally discuss how similar phenomena take place with the biologically more plausible mechanism of eliminating signals which are locally not coherent with the majority.