Cyclic Flats of a Polymatroid

Abstract

Polymatroids can be considered as “fractional matroids” where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to polymatroids. Defining cyclic flats of a polymatroid carefully, the characterization by Bonin and de Mier of the ranked lattice of cyclic flats carries over to polymatroids. The main tool, which might be of independent interest, is a convolution-like method which creates a polymatroid from a ranked lattice and a discrete measure. Examples show the ease of using the convolution technique.

The Cyclic Flats of a $q$-Matroid

“…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. Motivated by these questions and by the work of Csirmaz [10] on cyclic flats of classical polymatroids, we considered the question of generalizing the cryptomorphism from Theorem 3.5 to q-polymatroids.…”

“…This was shown in relation to distributed data storage; see [12,26]. Furthermore, even the lattice of cyclic flats of classical polymatroids has been considered; see [10,27,28].…”

The cyclic flats of a q-matroid

“…where the last equality follows from the assumption that g ∈ C U . Since C is nondegenerate, from (11) we have that ag t = 0 and so we arrive at a contradiction. This shows that the C V = C U for every subspace U ∈ Hyp(V ) and, in other words, the support of every codeword in the dual code of C is a cyclic space in the matroid M. The lattice of flats of M C ⊥ can be found in Fig.…”

“…q-Polymatroids and their connections to rank-metric codes were introduced in [15] and [23]. In [11], it was shown that knowledge of the lattice of cyclic flats, along with the ranks of its elements, is sufficient to determine a polymatroid. It is a natural question to ask whether or not a q-analogue of this result holds.…”