  Let $G$ be a simple connected graph. For any two vertices $u$ and $v$, let
$d(u,v)$ denote the distance between $u$ and $v$ in $G$. A radio-$k$-labeling
of $G$ for a fixed positive integer $k$ is a function $f$ which assigns to each
vertex a non-negative integer label such that for every two vertices $u$ and $
v $ in $G$, $|f(u)-f(v)| \geq k - d(u,v) +1$. The span of $f$ is the difference
between the largest and smallest labels of $f(V)$. The radio-$k$-number of a
graph $G$, denoted by $rn_k(G)$, is the smallest span among all
radio-$k$-labelings admitted by $G$. A cycle $C_n$ has diameter $d=\lfloor n/2
\rfloor$. In this paper, we combine a lower bound approach with cyclic group
structure to determine the value of $ rn_k(C_n)$ for $k \geq n-3$. For $d \leq
k < n-3$, we obtain the values of $rn_k(C_n)$ when $n$ and $k$ have the same
parity, and prove partial results when $n$ and $k$ have different parities. Our
results extend the known values of $rn_d (C_n)$ and $rn_{d+1} (C_n)$ shown by
Liu and Zhu, and by Karst, Langowitz, Oehrlein and Troxell, respectively.
