Bounding minimum distances of cyclic codes using algebraic geometry, by Nigel Boston
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
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.
Nigel Boston <email@example.com>