MacWilliams' Extension Theorem for rank-metric codes
Information Theory
2023-04-27 v1 math.IT
Abstract
The MacWilliams' Extension Theorem is a classical result by Florence Jessie MacWilliams. It shows that every linear isometry between linear block-codes endowed with the Hamming distance can be extended to a linear isometry of the ambient space. Such an extension fails to exist in general for rank-metric codes, that is, one can easily find examples of linear isometries between rank-metric codes which cannot be extended to linear isometries of the ambient space. In this paper, we explore to what extent a MacWilliams' Extension Theorem may hold for rank-metric codes. We provide an extensive list of examples of obstructions to the existence of an extension, as well as a positive result.
Cite
@article{arxiv.2304.13341,
title = {MacWilliams' Extension Theorem for rank-metric codes},
author = {Elisa Gorla and Flavio Salizzoni},
journal= {arXiv preprint arXiv:2304.13341},
year = {2023}
}
Comments
12 pages