Exceptional full-block MSRD systems with q blocks

Construct, or prove impossible, an MSRD system of dimension m+1 over the field of order q^m with profile (m,…,m) consisting of q blocks, for q≥3 and m≥2; equivalently, determine whether such a system exists in dimension (q−1)m−1, and in particular whether a three-dimensional MSRD code over the field of order q^2 can have q blocks of length 2.

Background

For MSRD systems whose blocks all have full length m, the paper proves the upper bound t≤q−1 when the smaller of the dimension and codimension satisfies κ≤m. When κ=m+1, the general bound permits the exceptional possibility t=q.

The paper shows that this possibility cannot occur for q=2, while an explicit construction later establishes a positive example for (q,m)=(3,2). The general existence question for q≥3 and m≥2 therefore remains unresolved.

References

Let $q\geq3$ and $m\geq2$. Is there an MSRD system of dimension $m+1$ over ${qm}$ with profile $(m,\ldots,m)$ consisting of $q$ blocks? Equivalently, by \cref{lem:puncture-dual}(ii), is there one of dimension $(q-1)m-1$? For $m=2$, this asks for a three-dimensional MSRD code over ${q2}$ with $q$ blocks of length $2$.

— On the Length of Maximum Sum-Rank Distance Codes  (2610.23633 - Astore et al., 16 Oct 2026) in Problem 4.12, Section 5.3, “The exceptional full-block case”

The preceding results provide several restrictions on the possible profiles of MSRD systems, but a general characterization of the admissible parameters remains open.

— On the Length of Maximum Sum-Rank Distance Codes  (2610.23633 - Astore et al., 16 Oct 2026) in Section 6, subsection “An MSRD conjecture”

Understanding whether further exceptions exist, particularly in the parameter range considered in \cref{prob:exceptional}, remains an interesting problem.

— On the Length of Maximum Sum-Rank Distance Codes  (2610.23633 - Astore et al., 16 Oct 2026) in Section 7, Conclusions