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.

翻译：我们用图形光谱法改进Gilbert-Varshamov。 Gilbert 图形 $Gq,n,d}$ 是一张图表,所有矢量都以$mathbb{F ⁇ q ⁇ n$作为顶点,如果两个顶点距离低于$d$,则两个顶点相邻。在本文中,我们使用 Cayley 图形的特性计算 $Gq,n,d}$的源值和源值。 改进的边点与该图形的最小 egen值相关。 最后我们给出一种算法, 来计算符合约束值的约束值和线性代码 。