Improving the Gilbert-Varshamov Bound by Graph Spectral Method

Abstract

We improve Gilbert-Varshamov bound by graph spectral method. Gilbert graph $G_{q,n,d}$ is a graph with all vectors in $\mathbb{F}^n_q$ 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.