On optimal nonlinear systematic codes

TL;DR

This paper investigates the applicability of the Griesmer bound to nonlinear systematic codes, identifying conditions under which it holds and providing explicit constructions where it does not.

Contribution

It characterizes when the Griesmer bound applies to nonlinear systematic codes and constructs examples where it fails, extending understanding of code optimality bounds.

Findings

The Griesmer bound holds for certain parameters in nonlinear systematic codes.

Explicit constructions show the bound does not always hold for systematic codes.

Some versions of the Griesmer bound are valid for all systematic codes.

Abstract

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.