Optimality of the Y(t,n,q) construction for even minimum distance

Determine whether the construction Y(t,n,q) of symmetric bilinear-form codes is optimal when the minimum distance d is even and strictly less than the ambient dimension n.

Background

Schmidt’s construction Y(t,n,q) produces symmetric rank-distance codes with prescribed parameters and was proved optimal when n=d and n is odd. The paper notes that the general case in which d is even and d<n had remained unresolved in the cited work. For the parameters studied in the present paper, namely n=3 and d=2, the authors observe that optimality can be established, but the broader parameter range remains the stated open question.

References

It was left as an open question whether this construction is optimal when d is even and d < n.

— Linear complete symmetric rank-distance codes  (2503.02586 - Alnajjarine et al., 4 Mar 2025) in Section 5, “Comparison with known constructions of MSRD codes”