Maps preserving a fixed rank-distance on matrices over finite fields

Finite Fields and Their Applications. 2026. Vol. 111. Article 102759.

Maksaev A., Medved N., Promyslov V.

Denote by Mm×n the space of all m × n matrices over a field. For a fixed 1 ⩽ k ⩽ min(m, n), we investigate bijective maps φ1, φ2 : Mm×n → Mm×n such that rk(A − B) = k iff rk(φ1(A) − φ2(B)) = k, for any A, B ∈ Mm×n. When k < min(m, n)/2, we not only characterize such maps on matrix spaces, but prove that such maps are equal isometries even on more general metric spaces that we call discrete-triangular. For an arbitrary k, we prove that the same characterization holds for the matrices over finite fields, except for 2 × 2 matrices over the field of 2 elements. To do this, we use theory of association schemes, specifically the bilinear forms scheme, and investigate its eigenvalues and intersection numbers

Keywords: Adjacency preserving mappings