On zero divisor graphs of finite commutative local rings

We describe the zero divisor graph of a commutative finite local rings $R$ of characteristic $2$ with Jacobson radical $J$ such that ${\dim_F J/J^2=2}$, ${\dim_F J^2/J^3=2}$, ${\dim_F J^3=1}$, $J^4=(0)$ and $F=R/J\cong GF(2^r)$, the finite field of $2^r$ elements.