  It has been demonstrated that modifying the rim scattering of a paraboloidal
reflector antenna through the use of reconfigurable elements along the rim
facilitates sidelobe modification including cancelling sidelobes. In this work
we investigate several open questions with respect to algorithms for
determining the weights. First, we derive the general weight values needed at
each reconfigurable element to place nulls at arbitrary angles. Second, since
in many cases these weights require gains other than one, we develop a
technique for determining unit-modulus weights so as to allow for surfaces
which merely modify the phase of the scattered field, while substantially
reducing the gain at arbitrary angles. Specifically, it is shown that despite
the large search space (and non-convexity in the presence of discrete weights),
weights can be found with reasonable computational complexity which provide
useful cancellation capability. Third, it is demonstrated that this can be done
using open-loop (i.e., with pattern knowledge), closed-loop (without pattern
knowledge), or hybrid (with inexact pattern knowledge) techniques both
quantized and unquantized phases. A primary finding is that sufficiently deep
nulls are possible with essentially no change in the main lobe gain with
practical (binary or quaternary) phase-only weights.
