Journal
JOURNAL OF SYMBOLIC COMPUTATION
Volume 122, Issue -, Pages -Publisher
ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
DOI: 10.1016/j.jsc.2023.102263
Keywords
Rank-metric codes; Isometries; MacWilliams' Extension Theorem
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The MacWilliams' Extension Theorem, proposed by Florence Jessie MacWilliams, investigates the extension of linear isometries between linear block-codes to linear isometries of the ambient space. This paper explores the applicability of this theorem to rank-metric codes, providing examples and results.
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. (c) 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).
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