q-polymatroids and their relation to rank-metric codes

Gluesing-Luerssen, Heide; Jany, Benjamin

doi:10.1007/s10801-022-01129-y

Cited by 13 publications

(16 citation statements)

References 17 publications

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“…M * \ V ∼ = (M/V ) * (equality rather than equivalence holds for matroids only). A proof of this fact for q-matroids can be found in [9,Thm 5.3] and in [15,Sect. 3] for matroids.…”

Section: Basic Notions Of Matroids and Q-matroidsmentioning

confidence: 92%

“…It was shown in [9,Thm 2.8] that given two NSBFs •, • 1 , •, • 2 , the respective dual q-matroids M * 1 and M * 2 of M are equivalent. For both matroids and q-matroids, an element v of the groundset, respectively groundspace, is a coloop if e is loop in the dual matroid, respectively dual q-matroid.…”

Section: Basic Notions Of Matroids and Q-matroidsmentioning

confidence: 99%

“…They were first studied by Jurrius and Pellikaan in [14], who showed that an F q m -linear rank metric code induces a q-matroid. It was shown later on that matrix linear rank metric codes induce a q-polymatroid, a generalization of q-matroids (see [9,10]). Since then, many results on q-(poly)matroids, and how they relate to rank metric codes, have been established, see for example [2,5,6,8,9,10,13].…”

Section: Introductionmentioning

confidence: 99%

“…It was shown later on that matrix linear rank metric codes induce a q-polymatroid, a generalization of q-matroids (see [9,10]). Since then, many results on q-(poly)matroids, and how they relate to rank metric codes, have been established, see for example [2,5,6,8,9,10,13]. Because of their q-analogue nature, many of the newly discovered properties of q-matroids turn out to be analogues of well established matroid theory results.…”

Section: Introductionmentioning

confidence: 99%

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The Projectivization Matroid of a $q$-Matroid

Jany¹

2022

Preprint

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In this paper, we investigate the relation between a q-matroid and its associated matroid called the projectivization matroid. The latter arises by projectivizing the groundspace of the q-matroid and considering the projective space as the groundset of the associated matroid on which is defined a rank function compatible with that of the q-matroid. We show that the projectivization map is a functor from categories of q-matroids to categories of matroids, which allows to prove new results about maps of q-matroids. We furthermore show the characteristic polynomial of a q-matroid is equal to that of the projectivization matroid. We use this relation to establish a recursive formula for the characteristic polynomial of a q-matroid in terms of the characteristic polynomial of its minors. Finally we use the projectivization matroid to prove a q-analogue of the critical theorem in terms of F q m -linear rank metric codes and q-matroids.

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“…M * \ V ∼ = (M/V ) * (equality rather than equivalence holds for matroids only). A proof of this fact for q-matroids can be found in [9,Thm 5.3] and in [15,Sect. 3] for matroids.…”

Section: Basic Notions Of Matroids and Q-matroidsmentioning

confidence: 92%

Section: Basic Notions Of Matroids and Q-matroidsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Projectivization Matroid of a $q$-Matroid

Jany¹

2022

Preprint

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“…Thanks to their relation to rank-metric codes, q-matroids and q-polymatroids have recently garnered a lot of attention, [1,2,3,4,5,6,7,10]. Indeed, F q m -linear rankmetric codes in F n q m give rise to q-matroids, whereas F q -linear rank-metric codes induce q-polymatroids.…”

Section: Introductionmentioning

confidence: 99%

Independent Spaces of q-Polymatroids

Gluesing-Luerssen¹,

Jany²

2022

Algebraic Combinatorics

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This paper is devoted to the study of independent spaces of q-polymatroids. With the aid of an auxiliary q-matroid it is shown that the collection of independent spaces satisfies the same properties as for q-matroids. However, in contrast to q-matroids, the rank value of an independent space does not agree with its dimension. Nonetheless, the rank values of the independent spaces fully determine the q-polymatroid, and this fact can be exploited to derive a cryptomorphism of q-polymatroids. Finally, the notions of minimal spanning spaces, maximally strongly independent spaces, and bases will be elaborated on.

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The direct sum of q-matroids

Ceria,

Jurrius

2024

J Algebr Comb

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For classical matroids, the direct sum is one of the most straightforward methods to make a new matroid out of existing ones. This paper defines a direct sum for q-matroids, the q-analogue of matroids. This is a lot less straightforward than in the classical case, as we will try to convince the reader. With the use of submodular functions and the q-analogue of matroid union we come to a definition of the direct sum of q-matroids. As a motivation for this definition, we show it has some desirable properties.

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q-polymatroids and their relation to rank-metric codes

Cited by 13 publications

References 17 publications

The Projectivization Matroid of a $q$-Matroid

The Projectivization Matroid of a $q$-Matroid

Independent Spaces of q-Polymatroids

The direct sum of q-matroids

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