  We give an independent set of size $367$ in the fifth strong product power of
$C_7$, where $C_7$ is the cycle on $7$ vertices. This leads to an improved
lower bound on the Shannon capacity of $C_7$: $\Theta(C_7)\geq 367^{1/5} >
3.2578$. The independent set is found by computer, using the fact that the set
$\{t \cdot (1,7,7^2,7^3,7^4) \,\, | \,\, t \in \mathbb{Z}_{382}\} \subseteq
\mathbb{Z}_{382}^5$ is independent in the fifth strong product power of the
circular graph $C_{108,382}$. Here the circular graph $C_{k,n}$ is the graph
with vertex set $\mathbb{Z}_{n}$, the cyclic group of order $n$, in which two
distinct vertices are adjacent if and only if their distance (mod $n$) is
strictly less than $k$.
