  Most bounds on the size of codes hold for any code, whether linear or not.
Notably, the Griesmer bound holds only in the linear case and so optimal linear
codes are not necessarily optimal codes. In this paper we identify code
parameters $(q,d,k)$, namely field size, minimum distance and dimension, for
which the Griesmer bound holds also in the (systematic) nonlinear case.
Moreover, we show that the Griesmer bound does not necessarily hold for a
systematic code by explicit construction of a family of optimal systematic
binary codes. On the other hand, we are able to provide some versions of the
Griesmer bound holding for all systematic codes.
