2020
DOI: 10.48550/arxiv.2005.03369
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Constructions of new matroids and designs over GF(q)

Abstract: A perfect matroid design (PMD) is a matroid whose flats of the same rank all have the same size. In this paper we introduce the q-analogue of a PMD and its properties. In order to do that, we first establish new cryptomorphic definitions for q-matroids. We show that q-Steiner systems are examples of q-PMD's and we use this matroid structure to construct subspace designs from q-Steiner systems. We apply this construction to S(2, 13, 3; q) Steiner systems and hence establish the existence of subspace designs wit… Show more

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Cited by 4 publications
(8 citation statements)
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“…The concept of q-matroid may be traced back to Crapo's PhD thesis [9]. More recently, the relation between rank-metric codes and q-matroids has led to these combinatorial objects getting a lot of attention from researchers; see for instance [5][6][7][8][13][14][15][16][17]20,24]. Indeed, it is wellknown that q-matroids generalize F q m -linear rank-metric codes, as classical matroids generalize linear codes in the Hamming metric.…”
Section: Introductionmentioning
confidence: 99%
“…The concept of q-matroid may be traced back to Crapo's PhD thesis [9]. More recently, the relation between rank-metric codes and q-matroids has led to these combinatorial objects getting a lot of attention from researchers; see for instance [5][6][7][8][13][14][15][16][17]20,24]. Indeed, it is wellknown that q-matroids generalize F q m -linear rank-metric codes, as classical matroids generalize linear codes in the Hamming metric.…”
Section: Introductionmentioning
confidence: 99%
“…They generalize the analogous notions for classical matroids and q-matroids. While for q-matroids the lattice of flats is semimodular [3], we will see that this is not the case for q-polymatroids. Nonetheless, if the q-polymatroid arises from a rank-metric code, the collection of flats turns out to be closely related to the code: the generalized weights of the code are fully determined by the flats.…”
Section: Introductionmentioning
confidence: 77%
“…Furthermore, Jurrius/Pellikaan present various cryptomorphic definitions of q-matroids. In the very recent preprint [4] (and the precursor [3]), Byrne and co-authors considerably extend the list of cryptomorphic definitions. It should be noted that q-matroids appeared already much earlier in the Ph.D. thesis [6] but remained unnoticed in the coding community until [13].…”
Section: Introductionmentioning
confidence: 99%
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“…We return to general q-matroids. In [2] (and partly in [1]) a variety of cryptomorphic definition of q-matroids have been derived. We will need the one based on flats.…”
Section: Introductionmentioning
confidence: 99%