  This paper studies circular designs for interference models, where a
treatment assigned to a plot also affects its neighboring plots within a block.
For the purpose of estimating total effects, the circular neighbor balanced
design was shown to be universally optimal among designs which do not allow
treatments to be neighbors of themselves. Our study shows that self-neighboring
block sequences are actually the main ingredient for an optimal design. Here,
we adopt the approximate design framework and study optimal designs in the
whole design space. Our approach is flexible enough to accommodate all possible
design parameters, that is the block size and the number of blocks and
treatments. This approach can be broken down into two main steps: the
identification of the minimal supporting set of block sequences and the
optimality condition built on it. The former is critical for reducing the
computational time from almost infinity to seconds. Meanwhile, the task of
finding the minimal set is normally achieved through numerical methods, which
can only handle small block sizes. Our approach is of a hybrid nature in order
to deal with all design sizes. When block size is not large, we provide
explicit expressions of the minimal set instead of relying on numerical
methods. For larger block sizes when a typical numerical method would fail, we
theoretically derived a reasonable size intermediate set of sequences, from
which the minimal set can be quickly derived through a customized algorithm.
Taking it further, the optimality conditions allow us to obtain both symmetric
and asymmetric designs. Lastly, we also investigate the trade-off issue between
circular and noncircular designs, and provide guidelines on the choices.
