Recoverable Systems

TL;DR

This paper investigates the maximum growth rate of recoverable sequence sets with unique local recoverability, connecting it to constrained systems, and extends the analysis to probabilistic models using ergodic theory.

Contribution

It introduces a novel recoverability framework for infinite sequences, establishes capacity bounds, and explores entropy-maximizing measures under probabilistic recoverability constraints.

Findings

Established the capacity bounds for recoverable sequence sets.

Connected the problem to constrained systems and ergodic theory.

Proposed methods to construct entropy-maximizing measures.

Abstract

Motivated by the established notion of storage codes, we consider sets of infinite sequences over a finite alphabet such that every $k$-tuple of consecutive entries is uniquely recoverable from its $l$-neighborhood in the sequence. We address the problem of finding the maximum growth rate of the set, which we term capacity, as well as constructions of explicit families that approach the optimal rate. The techniques that we employ rely on the connection of this problem with constrained systems. In the second part of the paper we consider a modification of the problem wherein the entries in the sequence are viewed as random variables over a finite alphabet that follow some joint distribution, and the recovery condition requires that the Shannon entropy of the $k$-tuple conditioned on its $l$-neighborhood be bounded above by some $\epsilon>0.$ We study properties of measures on infinite sequences that maximize the metric entropy under the recoverability condition. Drawing on tools from ergodic theory, we prove some properties of entropy-maximizing measures. We also suggest a procedure of constructing an $\epsilon$-recoverable measure from a corresponding deterministic system.