  Most bounds on the size of codes hold for any code, whether linear or
nonlinear. Notably, the Griesmer bound, holds only in the linear case. In this
paper we characterize a family of systematic nonlinear codes for which the
Griesmer bound holds. Moreover, we show that the Griesmer bound does not
necessarily hold for a systematic code by showing explicit counterexamples. On
the other hand, we are also able to provide (weaker) versions of the Griesmer
bound holding for all systematic codes.
