On extendibility of additive code isometries

TL;DR

This paper investigates when additive code isometries can be extended to the entire space, establishing conditions under which an analogue of the MacWilliams Extension Theorem applies for certain code lengths.

Contribution

It proves that for additive codes below a certain length threshold, all isometries extend to the whole space, and describes a family of unextendible isometries at that threshold.

Findings

Extension of additive code isometries depends on code length

An analogue of MacWilliams Extension Theorem holds below a certain length

A family of unextendible isometries is characterized at the threshold length

Abstract

For linear codes, the MacWilliams Extension Theorem states that each linear isometry of a linear code extends to a linear isometry of the whole space. But, in general, it is not the situation for nonlinear codes. In this paper it is proved, that if the length of an additive code is less than some threshold value, then an analogue of the MacWilliams Extension Theorem holds. One family of unextendible code isometries for the threshold value of code length is described.