A proof of Ward's conjecture on Griesmer codes

Let $C$ be an $[n,k,d{]}_q$ Griesmer code, where $q=p^f$ with $p$ prime, and write $e={\nu }_p(d)$. Ward conjectured that $C$ is $p^{\max \{0,e-f+1\}}$-divisible. In this paper, we prove this conjecture.