Greismer Codes with Maximum Covering Radius

Let $C$ be a $(k, d)$ Greismer code with covering radius $\rho(C)$. We prove that if $d>2^k$ or d belongs to Belov intervals of dimension $k+1$, then $\rho(C)\leq d-2$. All Greismer codes $C$ with $\rho(C)\leq d-1$ are described.