Independent Spaces of q-Polymatroids

Abstract

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.

The Cyclic Flats of a $q$-Matroid

“…q-Polymatroids and their connection to rank-metric codes were introduced in [17] and [24]. Their properties have been further developed in [5,14,15]. It is not the purpose of this paper to discuss q-polymatroid in general, hence, we only propose the definition for the sake of completeness.…”

“…Contrary to the q-matroid case, for q-polymatroids there are not many cryptomorphisms. In [15] a cryptomorphic definition of q-polymatroids in terms of independent spaces along with their rank values has been derived. Moreover, in [14], the authors provided examples showing that the rank values of the flats need to be taken into account in order determine a q-matroid from its lattice of flats.…”

“…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.…”