Robust Positioning Patterns with Low Redundancy

TL;DR

This paper introduces new binary and q-ary positioning patterns and sequences that are highly robust to errors, with near-optimal redundancy and efficient locating algorithms, advancing the reliability of positioning systems.

Contribution

It provides the first explicit binary positioning patterns and arrays with robustness to errors, achieving near-optimal redundancy and efficient locating algorithms.

Findings

Binary patterns robust to constant errors with minimal redundancy.

Explicit constructions of q-ary sequences with high error tolerance.

Locating algorithms run in cubic time relative to sequence length or array dimension.

Abstract

A robust positioning pattern is a large array that allows a mobile device to locate its position by reading a possibly corrupted small window around it. In this paper, we provide constructions of binary positioning patterns, equipped with efficient locating algorithms, that are robust to a constant number of errors and have redundancy within a constant factor of optimality. Furthermore, we modify our constructions to correct rank errors and obtain binary positioning patterns robust to any errors of rank less than a constant number. Additionally, we construct $q$-ary robust positioning sequences robust to a large number of errors, some of which have length attaining the upper bound. Our construction of binary positioning sequences that are robust to a constant number of errors has the least known redundancy amongst those explicit constructions with efficient locating algorithms. On the other hand, for binary robust positioning arrays, our construction is the first explicit construction whose redundancy is within a constant factor of optimality. The locating algorithms accompanying both constructions run in time cubic in sequence length or array dimension.