Wei-type duality theorems for rank metric codes

A polymatroid approach to generalized weights of rank metric codes

“…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). For the other inequality, it suffices to consider X 0 ∈ (E) with max{dim C(X 0 ), dim C T (X 0 )} ≥ r such that d r (P) = dim X 0 .…”

“…(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. Thus, ρ * satisfies (R1).…”

“…After this work was submitted and put on the arXiv, we learned of the work of Britz et al [5] where results similar to those in Sect. 3 of this paper are proved, albeit using different methods.…”

“…3 of this paper are proved, albeit using different methods. A nice comparison of the two approaches is given in the postscript of [5] and we refer the interested reader to it.…”

Shellability and homology of $q$-complexes and $q$-matroids

“…These are obtained by looking at a minimal graded free resolution of the Stanley-Reisner ring of a simplicial complex that corresponds to the vector matroid associated to the parity check matrix of the given linear code. The question that arises naturally is whether something like Betti numbers can be defined in the context of rank metric codes, or more generally, for q-matroids as in [10] or going even further, for the pq, mq-polymatroids studied in [15,6,8]. We were led to the study of shellability and homology of q-complexes, and especially, complexes associated to q-matroids with a view toward a possible topological approach to the above question.…”