  The Shannon capacity of a graph $G$ is defined as $c(G)=\sup_{d\geq
1}(\alpha(G^d))^{\frac{1}{d}},$ where $\alpha(G)$ is the independence number of
$G$. The Shannon capacity of the cycle $C_5$ on $5$ vertices was determined by
Lov\'{a}sz in 1979, but the Shannon capacity of a cycle $C_p$ for general odd
$p$ remains one of the most notorious open problems in information theory. By
prescribing stabilizers for the independent sets in $C_p^d$ and using
stochastic search methods, we show that $\alpha(C_7^5)\geq 350$,
$\alpha(C_{11}^4)\geq 748$, $\alpha(C_{13}^4)\geq 1534$ and
$\alpha(C_{15}^3)\geq 381$. This leads to improved lower bounds on the Shannon
capacity of $C_7$ and $C_{15}$: $c(C_7)\geq 350^{\frac{1}{5}}> 3.2271$ and
$c(C_{15})\geq 381^{\frac{1}{3}}> 7.2495$.
