On the Minimum Length of Linear Codes over the Field of 9 Elements

We construct a lot of new $[n,4,d]_9$ codes whose lengths are close to the Griesmer bound and prove the nonexistence of some linear codes attaining the Griesmer bound using some geometric techniques through projective geometries to determine the exact value of $n_9(4,d)$ or to improve the known bound on $n_9(4,d)$ for given values of $d$, where $n_q(k,d)$ is the minimum length $n$ for which an $[n,k,d]_q$ code exists. We also give the updated table for $n_9(4,d)$ for all $d$ except some known cases.