q-polymatroids and their relation to rank-metric codes

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Abstract

It is well known that linear rank-metric codes give rise to q-polymatroids. Analogously to matroid theory, one may ask whether a given q-polymatroid is representable by a rank-metric code. We provide an answer by presenting an example of a q-matroid that is not representable by any linear rank-metric code and, via a relation to paving matroids, provide examples of various q-matroids that are not representable by Fqm-linear rank-metric codes. We then go on and introduce deletion and contraction for q-polymatroids and show that they are mutually dual and correspond to puncturing and shortening of rank-metric codes. Finally, we introduce a closure operator along with the notion of flats and show that the generalized rank weights of a rank-metric code are fully determined by the flats of the associated q-polymatroid.

Original languageEnglish
Pages (from-to)725-753
Number of pages29
JournalJournal of Algebraic Combinatorics
Volume56
Issue number3
DOIs
StatePublished - Nov 2022

Bibliographical note

Funding Information:
Heide Gluesing-Luerssen, HGL was partially supported by the Grant #422479 from the Simons Foundation.

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Rank-metric codes
  • Representability
  • q-matroids
  • q-polymatroids

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics

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