The Shannon capacity of a graph $G$ is defined as $c(G)=\sup_{d\geq 1}(α(G^d))^{\frac{1}{d}},$ where $α(G)$ is the independence number of $G$. The Shannon capacity of the cycle $C_5$ on $5$ vertices was determined by Lovász 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 $α(C_7^5)\geq 350$, $α(C_{11}^4)\geq 748$, $α(C_{13}^4)\geq 1534$ and $α(C_{15}^3)\geq 381$. 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 $c(C_{15})\geq 381^{\frac{1}{3}}> 7.2495$.