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no. 4
Independent Spaces of q-Polymatroids
Gluesing-Luerssen, Heide1; Jany, Benjamin1
1 University of Kentucky Dept. of Mathematics Lexington, KY 40506 (USA)
Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 727-744.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.241
Classification: 05B35, 05A30
Keywords: $q$-polymatroids, rank-metric codes, independent spaces.

Gluesing-Luerssen, Heide 1; Jany, Benjamin 1

1 University of Kentucky Dept. of Mathematics Lexington, KY 40506 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Gluesing-Luerssen, Heide; Jany, Benjamin. Independent Spaces of $q$-Polymatroids. Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 727-744. doi : 10.5802/alco.241. https://alco.centre-mersenne.org/articles/10.5802/alco.241/

[1] Byrne, E.; Ceria, M.; Ionica, S.; Jurrius, R. Weighted Subspace Design from q-Polymatroids (2021) (preprint) | arXiv

[2] Byrne, E.; Ceria, M.; Ionica, S.; Jurrius, R.; Saçikara, E. Constructions of new matroids and designs over GF(q) (2020) (preprint) | arXiv

[3] Byrne, E.; Ceria, M.; Jurrius, R. Constructions of new q-cryptomorphisms, J. Comb. Theory. Ser. B, Volume 153 (2022), pp. 149-194 | DOI | MR | Zbl

[4] Ghorpade, S. R.; Johnsen, T. A polymatroid approach to generalized weights of rank metric codes, Des. Codes Cryptogr., Volume 88 (2020), pp. 2531-2546 | DOI | MR | Zbl

[5] Gluesing-Luerssen, Heide; Jany, Benjamin q-Polymatroids and their Relation to Rank-Metric Codes (2021) (preprint, accepted for publication in J. Algebraic Combin.) | arXiv

[6] Gorla, E.; Jurrius, R.; López, H.; Ravagnani, A. Rank-metric codes and q-polymatroids, J. Algebraic Combin., Volume 52 (2020), pp. 1-19 | DOI | MR | Zbl

[7] Jurrius, R.; Pellikaan, R. Defining the q-analogue of a matroid, Electron. J. Combin., Volume 25 (2018) no. 3, Paper no. 3.2, 32 pages | DOI | MR | Zbl

[8] Oxley, J. Matroid Theory, Oxford Graduate Text in Mathematics, Oxford University Press, 2011 | DOI

[9] Ravagnani, A. Rank-metric codes and their duality theory, Des. Codes Cryptogr., Volume 80 (2016), pp. 197-216 | DOI | MR | Zbl

[10] Shiromoto, K. Codes with the rank metric and matroids, Des. Codes Cryptogr., Volume 87 (2019), pp. 1765-1776 | DOI | MR | Zbl

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