MacWilliams Extension Theorem for MDS additive codes

TL;DR

This paper proves an analogue of the MacWilliams Extension Theorem for additive MDS codes, showing that most additive isometries extend to ambient space isometries, unlike the general case for additive codes.

Contribution

It establishes that for almost all additive MDS codes, additive isometries extend to ambient space isometries, filling a gap in the theory of additive codes.

Findings

Additive isometries extend to ambient space for almost all additive MDS codes.

The extension property holds in the additive case for MDS codes, unlike general additive codes.

The result generalizes the classical MacWilliams Extension Theorem to a broader class of codes.

Abstract

The MacWilliams Extension Theorem states that each linear isometry of a linear code extends to a monomial map. Unlike the linear codes, in general, additive codes do not have the extension property. In this paper, an analogue of the extension theorem for additive codes in the case of additive MDS codes is proved. More precisely, it is shown that for almost all additive MDS codes their additive isometries extend to isometries of the ambient space.