q-Polymatroids and Their Relation to Rank-Metric Codes

Abstract

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.

Coproducts in Categories of q-Matroids

“…Not every q-matroid is representable. The first non-representable q-matroid appeared in [6,Ex. 4.9].…”

“…Define ρ(V ) = 1 for V ∈ V and ρ(V ) = min{2, dim V } otherwise. It follows from [6,Prop. 4.7] that this does indeed define a q-matroid M = (F 4 2 , ρ).…”

“…In general, this leads to a q-polymatroid, that is, some of the rank values are not integers (see [7,Thm. 5.3] or [6]). If the code C is F q m -linear (in the sense of [6,Def.…”

“…5.3] or [6]). If the code C is F q m -linear (in the sense of [6,Def. 3.6]), then ρ C is integer-valued and thus defines a q-matroid M C = (F n , ρ C ).…”

Independent Spaces of q-Polymatroids

“…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 rank-metric codes in F n q m give rise to qmatroids, whereas F q -linear rank-metric codes induce q-polymatroids.…”