On the Griesmer bound for nonlinear codes

TL;DR

This paper investigates the applicability of the Griesmer bound to nonlinear codes, identifying conditions under which it holds or fails, and providing counterexamples and weaker bounds for systematic codes.

Contribution

It characterizes when the Griesmer bound applies to systematic nonlinear codes and presents counterexamples where it does not, along with weaker alternative bounds.

Findings

The Griesmer bound holds for a specific family of systematic nonlinear codes.

Counterexamples show the bound does not always apply to systematic codes.

Weaker versions of the Griesmer bound are established for all systematic 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.