  We improve Gilbert-Varshamov bound by graph spectral method. Gilbert graph
$G_{q,n,d}$ is a graph with all vectors in $\mathbb{F}_q^n$ as vertices where
two vertices are adjacent if their Hamming distance is less than $d$. In this
paper, we calculate the eigenvalues and eigenvectors of $G_{q,n,d}$ using the
properties of Cayley graph. The improved bound is associated with the minimum
eigenvalue of the graph. Finally we give an algorithm to calculate the bound
and linear codes which satisfy the bound.
