Linearly Bounded Liars, Adaptive Covering Codes, and Deterministic Random Walks

TL;DR

This paper introduces the 'liar machine', a deterministic analogue of random walk, providing tight bounds and improving strategies in liar games and covering codes with specific parameters.

Contribution

It develops the liar machine model, yielding asymptotically tight bounds and new results for liar games and adaptive covering codes.

Findings

Established tight discrepancy bounds for the liar machine.

Improved winning strategies for the binary liar game.

Proved existence of efficient covering codes with near-optimal size.

Abstract

We analyze a deterministic form of the random walk on the integer line called the {\em liar machine}, similar to the rotor-router model, finding asymptotically tight pointwise and interval discrepancy bounds versus random walk. This provides an improvement in the best-known winning strategies in the binary symmetric pathological liar game with a linear fraction of responses allowed to be lies. Equivalently, this proves the existence of adaptive binary block covering codes with block length $n$, covering radius $\leq fn$ for $f\in(0,1/2)$, and cardinality $O(\sqrt{\log \log n}/(1-2f))$ times the sphere bound $2^n/\binom{n}{\leq \lfloor fn\rfloor}$.