What question did this study set out to answer?

The aim is to establish the dimension of the vector space generated by the characteristic vectors of MRD codes.

April 25, 2026

On the Dimension of the Space Generated by Characteristic Vectors of MRD Codes

Key Points

The aim is to establish the dimension of the vector space generated by the characteristic vectors of MRD codes.
Defined the space generated by MRD codes using characteristic vectors.
Proved the relationship between the dimension and the valency of bilinear forms.
Analyzed matrices of a specific rank to determine their contribution to the vector space.
Proved that the dimension of the space equals the valency of the bilinear forms scheme.
Identified that this valency corresponds to the count of matrices of a certain rank.
Established a clear relationship between character vectors and MRD codes within rank-metric codes.

Abstract

ABSTRACT Given positive integers (), we call a rank‐metric code , with minimum distance for some , a maximum rank distance code (or MRD code for short) if . The space generated by MRD codes is defined to be the ‐vector space spanned by the characteristic vectors of all MRD codes. In this paper, we prove that its dimension equals , where the ‐th valency of the bilinear forms scheme, , is exactly the number of matrices of rank in .

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