  In 1980, R. Tijdeman provided an on-line algorithm that generates sequences
over a finite alphabet with minimal discrepancy, that is, such that the
occurrence of each letter optimally tracks its frequency. In this article, we
define discrete dynamical systems generating these sequences. The dynamical
systems are defined as exchanges of polytopal pieces, yielding cut and project
schemes, and they code tilings of the line whose sets of vertices form model
sets. We prove that these sequences of low discrepancy are natural codings of
toral translations with respect to polytopal atoms, and that they generate a
minimal and uniquely ergodic subshift with purely discrete spectrum. Finally,
we show that the factor complexity of these sequences is of polynomial growth
order $n^{d-1}$, where $d$ is the cardinality of the alphabet.
