On [formula omitted]-Steiner systems from rank metric codes

Abstract

In this paper we prove that rank metric codes with special properties imply the existence of q-analogs of suitable designs. More precisely, we show that the minimum weight vectors of a [2d,d,d] dually almost MRD code C≤Fqm2d(2d≤m) which has no code words of rank weight d+1 form a q-Steiner system S(d−1,d,2d)q. This is the q-analog of a result in classical coding theory and it may be seen as a first step to prove a q-analog of the famous Assmus–Mattson Theorem.