TY - JOUR
T1 - Improving the Gilbert-Varshamov Bound by Graph Spectral Method
AU - Ye , Zicheng
AU - Zhang , Huazi
AU - Li , Rong
AU - Wang , Jun
AU - Yan , Guiying
AU - Ma , Zhiming
JO - CSIAM Transactions on Applied Mathematics
VL - 1
SP - 1
EP - 12
PY - 2023
DA - 2023/01
SN - 4
DO - http://doi.org/ 10.4208/csiam-am.SO-2021-0024
UR - https://global-sci.org/intro/article_detail/csiam-am/21333.html
KW - Gilbert–Varshamov bound, independence number, graph spectral method, Cayley graph, linear codes.
AB - <p style="text-align: justify;">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.</p>