Existence of length 2m−3 rank-metric intersecting codes for odd m

Determine whether, for odd integers m and any prime power q, there exists a nondegenerate [2m−3,3,d]_{q^m/q} rank-metric intersecting code for some minimum distance d.

Background

The paper proves that the maximal length n=2m−3 can only occur when k=3 and m≥6, and gives a geometric characterization in terms of scattered Fq-subspaces of F_{qm}3 of dimension m+3. Using known constructions, existence is established when m is even. However, for odd m, the existence of sufficiently large scattered subspaces is not fully understood, leaving the corresponding codes unresolved.

Thus, despite the structural reductions and partial results, the existence of nondegenerate rank-metric intersecting codes with length 2m−3 remains an open question in the odd-m case.

References

In particular, when m is odd the existence of scattered subspaces of dimension at least m+3 in F_{qm}3 is not fully understood, and therefore the existence of rank-metric intersecting codes of length 2m-3 in this case remains open.

— On the existence of linear rank-metric intersecting codes  (2604.02004 - Borello et al., 2 Apr 2026) in Conclusions