On $q$-analog Steiner systems of rank metric codes

TL;DR

This paper establishes a connection between certain rank metric codes and the existence of $q$-analog Steiner systems, revealing that specific code properties imply the existence of these combinatorial designs.

Contribution

It proves that minimum weight vectors of particular rank metric codes form $q$-analog Steiner systems, linking coding theory with combinatorial design theory.

Findings

Minimum weight vectors form $q$-analog Steiner systems

Existence of such systems requires $d+1$ to be prime

Special properties of rank metric codes imply combinatorial designs

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 $C\leq \mathbb{F}_{q^m}^n$ which has no code words of rank weight $d+1$ form a $q$-analog Steiner system $S_q(d-1,d,2d)$. In particular, $d+1$ must be a prime.