Improving the Gilbert-Varshamov Bound by Graph Spectral Method

TL;DR

This paper enhances the Gilbert-Varshamov bound for coding theory by utilizing spectral graph theory, specifically eigenvalues of the Gilbert graph, leading to improved bounds and algorithms for linear codes.

Contribution

It introduces a spectral method to improve the Gilbert-Varshamov bound by analyzing eigenvalues of the Gilbert graph, providing a new approach to bound estimation.

Findings

Calculated eigenvalues and eigenvectors of Gilbert graph.

Derived an improved bound based on minimum eigenvalue.

Provided an algorithm for bound calculation and code construction.

Abstract

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