Robust Positioning Patterns with Low Redundancy

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.

Applications of combinatorics in coding theory

“…This combination is useful since substitution errors are prone to occur. In [6] and [2], several constructions for both binary RPSs and q-ary RPSs are proposed. The hard problem is these constructions contain markers, which currently contain a long run of zeros.…”

“…In this dissertation, we focus on binary sequences, i.e., onedimension arrays. In [6], we considered robust positioning patterns which are large arrays that allow a mobile device to locate its position by reading a possibly corrupted small 2D window around it. The design of such patterns is fundamental in robotics and has practical applications in areas, such as robot localization [51], camera localization [57], 3D surface imaging by structured light [22] and projected touchscreens [13].…”

“…Recently, Berkowitz and Kopparty [2] presented explicit constructions of binary (n, d)-RPS of length ⌦(2 n /n 9d ) when d is constant. In [6], the authors later constructed longer RPS in this regime, specifically, binary (n, d)-RPS of length ⌦(2 n /n 3d+6. 5 ).…”

“…Part of this chapter is presented in the 2020 IEEE International Symposium of Information Theory in [15] and appears SIAM Journal on Computing in [6].…”

“…Here we list optimal (n, d)-RPSs for n  13 and 2  d < b2n/3c, as well as (n, d) 2 {(4, 2), (7,4), (10,6)}. Trivialy, when d 2n/3, the maximum length of (n, d)-RPS is 2.…”