Rank-metric codes and q-polymatroids

Abstract

This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce q-polymatroids, the q-analogue of polymatroids, and develop their basic properties. We associate a pair of q-polymatroids to a rank-metric codes and show that several invariants and structural properties of the code, such as generalized weights, the property of being MRD or an optimal anticode, and duality, are captured by the associated combinatorial object.We start by establishing the notation and the definitions used throughout the paper.Notation 1.1. In the sequel, we fix integers n, m ≥ 2 and a prime power q. For an integer t, we let [t] := {1, ..., t}. We denote by F q the finite field with q elements. The space of n × m matrices with entries in F q is denoted by Mat. Up to transposition, we assume without loss of generality that n ≤ m. We let Mat(J, c) = {M ∈ Mat | colsp(M ) ⊆ J} and Mat(J, r) = {M ∈ Mat | rowsp(M ) ⊆ J}.Throughout the paper, we only consider linear codes. All dimensions are computed over F q , unless otherwise stated.

A polymatroid approach to generalized weights of rank metric codes

“…There is a natural connection between duals of Delsarte codes and the duals of (q, m)polymatroids. It is shown by Shiromoto [19] as well as Gorla et al [10], and we record it below.…”

“…This proves that P = (E, ρ) is a (q, m)demi-polymatroid. The desired formula for the conullity function of P is immediate from (10). This, in turn, shows that Indeed, the inequality d r (P) ≤ min{d r (P(C)), d r (P(C T ))} is clear from the definition and equation (5).…”

“…The proof is quite short and natural and given in [19,Proposition 11], and also in [10,Theorem 8.1]. An immediate consequence is the following.…”

“…It follows that ρ * (X ) = m dim X − max{dim C(X ), dim C T (X )}. (10) This implies that ρ * (X ) ≤ m dim X . Moreover, it also implies that ρ * (X ) ≥ 0, because from (5) and Proposition 6 we see that both m dim X − dim C(X ) and m dim X − dim C T (X ) are nonnegative.…”

“…Also, considering flags of length 1, we can deduce the results of Ravagnani [18] for the GW's of Delsarte codes and their duals. We remark that q-analogues of matroids, called q-matroids and q-polymatroids, have been considered by Jurrius and Pellikaan [12] and by Gorla et al [10], respectively. However, as far as we can see, Wei-type duality for their generalized weights is not shown in these papers.…”