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 n9(4, d) or to improve the known bound on n9(4, d) for given values of d, where nq(k, d) is the minimum length n for which an [n, k, d]q code exists. We also give the updated table for n9(4, d) for all d except some known cases.