  There are many results on the minimum distance of a cyclic code of the form
that if a certain set T is a subset of the defining set of the code, then the
minimum distance of the code is greater than some integer t. This includes the
BCH, Hartmann-Tzeng, Roos, and shift bounds and generalizations of these. In
this paper we define certain projective varieties V(T,t) whose properties
determine whether, if T is in the defining set, the code has minimum distance
exceeding t. Thus our attention shifts to the study of these varieties. By
investigating them using class field theory and arithmetical geometry, we will
prove various new bounds. It is interesting, however, to note that there are
cases that existing methods handle, that our methods do not, and vice versa. We
end with a number of conjectures.
