  Given a $q$-ary frequency hopping sequence set of length $n$ and size $M$
with Hamming correlation $H$, one can obtain a $q$-ary (nonlinear) cyclic code
of length $n$ and size $nM$ with Hamming distance $n-H$. Thus, every upper
bound on the size of a code from coding theory gives an upper bound on the size
of a frequency hopping sequence set. Indeed, all upper bounds from coding
theory have been converted to upper bounds on frequency hopping sequence sets
(\cite{Ding09}). On the other hand, a lower bound from coding theory does not
automatically produce a lower bound for frequency hopping sequence sets. In
particular, the most important lower bound--the Gilbert-Varshamov bound in
coding theory has not been transformed to frequency hopping sequence sets. The
purpose of this paper is to convert the Gilbert-Varshamov bound in coding
theory to frequency hopping sequence sets by establishing a connection between
a special family of cyclic codes (which are called hopping cyclic codes in this
paper) and frequency hopping sequence sets. We provide two proofs of the
Gilbert-Varshamov bound. One is based on probabilistic method that requires
advanced tool--martingale. This proof covers the whole rate region. The other
proof is purely elementary but only covers part of the rate region.
