The non-existence of Griesmer codes with parameters close to codes of Belov type

Hill and Kolev give a large class of q-ary linear codes meeting the Griesmer bound, which are called codes of Belov type (Hill and Kolev, Chapman Hall/CRC Research Notes in Mathematics 403, pp. 127---152, 1999). In this article, we prove that there are no linear codes meeting the Griesmer bound for values of d close to those for codes of Belov type. So we conclude that the lower bounds of d of codes of Belov type are sharp. We give a large class of length optimal codes with n q (k, d) = g q (k, d) + 1.