The BCH Family of Storage Codes on Triangle-Free Graphs and Its Relation to R(3, t)

Consider a simple, connected graph Γ with
n
vertices. Let
C
be a code of length
n
with its coordinates corresponding to the vertices of Γ. We define
C
as a
storage code
on Γ if, for any codeword
c
∈
C
, the information at each coordinate of
c
can be recovered by accessing its neighboring coordinates. The main problem here is to construct high-rate storage codes on triangle-free graphs. In this paper, we employ the polynomial method to address a question proposed by Barg and Zémor in 2022, demonstrating that the BCH family of storage codes on triangle-free Cayley graphs achieves a unit rate. Furthermore, we generalize the construction of the BCH family and obtain more storage codes of unit rate on triangle-free graphs. We also compare the BCH family with the other known constructions by examining the rate of convergence of 1/(1-
R
(
C_n
)) with respect to the length
n
, where
R
(
C_n
) is the rate of code
C_n
. At last, we reveal a connection between the storage codes on triangle-free graphs and the Ramsey number
R
(3,
t
), which leads to an upper bound for the rate of convergence of 1/(1 -
R
(
C_n
)).