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Rationality and computability of the covering radius for sofic shifts

Tom Meyerovitch, Aidan Young

Abstract

The covering radius of a shift space is a quantity of interest for information-theoretic applications of data transmission over noisy channels. We prove that the covering radius of a primitive sofic shift is a rational number, and describe an algorithm to compute the covering radius from a labeled graph presentation.

Rationality and computability of the covering radius for sofic shifts

Abstract

The covering radius of a shift space is a quantity of interest for information-theoretic applications of data transmission over noisy channels. We prove that the covering radius of a primitive sofic shift is a rational number, and describe an algorithm to compute the covering radius from a labeled graph presentation.
Paper Structure (10 sections, 20 theorems, 143 equations)

This paper contains 10 sections, 20 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Words and codes
  4. Labeled graphs
  5. Symbolic dynamics: Shift spaces and sofic shifts
  6. The covering radius of shift space
  7. Tropical convolution and maximal functions
  8. A zero sum two player mean payoff game
  9. The space of non-improvable paths
  10. Concluding remarks and open questions

Key Result

Theorem A

Let $\mathcal{G}=(G,L)$ be a primitive labeled graph. Then the covering radius $R(X_\mathcal{G})$ is a rational number.

Theorems & Definitions (40)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem C
  • ...and 30 more