A dimension bound for constant rank subspaces of matrices over a finite field

TL;DR

This paper establishes an upper bound on the dimension of constant rank subspaces of matrices over finite fields, showing they cannot exceed the number of columns when the field size is sufficiently large.

Contribution

It proves a new dimension bound for constant rank subspaces of matrices over finite fields, extending understanding of their structure and limitations.

Findings

Dimension of constant rank r subspaces is at most n.

Bound holds for finite fields with at least r+1 elements.

Results apply to matrices with rank r over finite fields.

Abstract

K be a field and let m and n be positive integers, where m does not exceed n. We say that a non-zero subspace of m x n matrices over K is a constant rank r subspace if each non-zero element of the subspace has rank r, where r is a positive integer that does not exceed m. We show in this paper that if K is a finite field containing at least r+1 elements, any constant rank r subspace of m x n matrices over K has dimension at most n.