Antipodal two-weight rank metric codes

TL;DR

This paper studies a special class of linear rank metric codes with two weights, providing their properties, constructions from MRD codes, and a complete classification in a specific case, also deriving related Hamming metric codes.

Contribution

It characterizes antipodal two-weight rank metric codes, proves their dimension is 2, and classifies them when the minimum rank distance equals half the length.

Findings

Dimension of such codes is 2.

Minimum rank distance is at least half of the length.

Complete classification when minimum rank distance equals half the length.

Abstract

We consider the class of linear antipodal two-weight rank metric codes and discuss their properties and characterization in terms of $t$-spreads. It is shown that the dimension of such codes is $2$ and the minimum rank distance is at least half of the length. We construct antipodal two-weight rank metric codes from certain MRD codes. A complete classification of such codes is obtained, when the minimum rank distance is equal to half of the length. As a consequence of our construction of two-weight rank metric codes, we get some explicit two-weight Hamming metric codes.