On $q$-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 [2$d$,$d$, $d$]dually almost MRD code $C \leq \mathbb{F}^{2d}_{g^m} (2d \leq m )$ which has no code words of rank weight $a + 1$ 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.