On the Griesmer bound for nonlinear codes

Abstract: 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.