q-Polymatroids and Their Relation to Rank-Metric Codes
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It is well known that linear rank-metric codes give rise to q-polymatroids. Analogously to classical matroid theory one may ask whether a given q-polymatroid is representable by a rank-metric code. We provide a partial answer by presenting examples of 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 that they 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.
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