q-polymatroids and their relation to rank-metric codes

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-me...

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Bibliographic Details
Published in:Journal of algebraic combinatorics 2022-11, Vol.56 (3), p.725-753
Main Authors: Gluesing-Luerssen, Heide, Jany, Benjamin
Format: Article
Language:English
Subjects:
Combinatorics
Computer Science
Convex and Discrete Geometry
Group Theory and Generalizations
Lattices
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
Piercing
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Summary: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 F q m -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.
ISSN:0925-9899
1572-9192
DOI:10.1007/s10801-022-01129-y