# On $q$-Steiner systems from rank metric codes

Arias, Francisco; de la Cruz, Javier; Rosenthal, Joachim; Willems, Wolfgang (2018). On $q$-Steiner systems from rank metric codes. Discrete Mathematics, 341(10):2729-2734.

## 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.

## 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.

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