A polymatroid approach to generalized weights of rank metric codes

Abstract

We consider the notion of a (q, m)-polymatroid, due to Shiromoto, and the more general notion of (q, m)-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality for generalized Hamming weights of linear codes. The corresponding results of Ravagnani for Delsarte rank metric codes, and Martínez-Peñas and Matsumoto for relative generalized rank weights are derived as a consequence.

q-Polymatroids and Their Relation to Rank-Metric Codes

“…While F q m -linear rank-metric codes give rise to q-matroids, this is not the case for general rankmetric codes. However, as shown by Gorla and co-authors in [12] as well as Shiromoto [19] and Ghorpade/Johnson [9], general rank-metric codes induce q-polymatroids. This means that the rank function attains rational values (for this reason polymatroids are also referred to as fractional matroids).…”

“…Then τ (V ) ∈ N 0 for all V ∈ V(F n ) and the map τ satisfies (R2) and (R3) from Definition 2.1 as well as τ (V ) ≤ µ dim(V ). Thus M ′ := (F n , τ ) is a (q, µ)-polymatroid in the sense of [9,Def. 1].…”

“…1]. In [9,Def. 10] the authors define the i-th generalized weight of M ′ as d i (M ′ ) = min{dim X | τ (F n ) − τ (X ⊥ ) ≥ i}.…”

“…In this paper we will make further contributions to the theory of q-polymatroids. Different from [12,9,19] we will study q-polymatroids over general n-dimensional F-vector spaces E rather than F n . This forces us to revisit a duality result from [12] and show that the equivalence class of the dual q-polymatroid does not depends on the choice of the non-degenerate symmetric bilinear form on E. The purpose of this slight generalization becomes clear only when we study deletions and contractions as the latter naturally lead to general ground spaces.…”

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.…”