On equivalence classes of matrices over a finite field of odd characteristic

In this article we classified up to isomorphism all finite local rings $R$ with Jacobson radical $J$ and conditions:
$$\mathrm{char} R\neq 2,\ R/J=F\subseteq Z(R),\ {\dim_F J/J^2=2},\ {\dim_F J^2=3},\ {J^3=0}.$$