  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.
