  We present several improvements to the Cauchy-characteristic evolution
procedure that generates high-fidelity gravitational waveforms at
$\mathcal{I}^+$ from numerical relativity simulations. Cauchy-characteristic
evolution combines an interior solution of the Einstein field equations based
on Cauchy slices with an exterior solution based on null slices that extend to
$\mathcal{I}^+$. The foundation of our improved algorithm is a comprehensive
method of handling the gauge transformations between the arbitrarily specified
coordinates of the interior Cauchy evolution and the unique (up to BMS
transformations) Bondi-Sachs coordinate system of the exterior characteristic
evolution. We present a reformulated set of characteristic evolution equations
better adapted to numerical implementation. In addition, we develop a method to
ensure that the angular coordinates used in the volume during the
characteristic evolution are asymptotically inertial. This provides a direct
route to an expanded set of waveform outputs and is guaranteed to avoid
pure-gauge logarithmic dependence that has caused trouble for previous spectral
implementations of the characteristic evolution equations. We construct a set
of Weyl scalars compatible with the Bondi-like coordinate systems used in
characteristic evolution, and determine simple, easily implemented forms for
the asymptotic Weyl scalars in our suggested set of coordinates.
