Characterization of when complete and maximum symmetric rank-distance codes coincide

Characterize the conditions under which every Fq-linear complete symmetric rank-distance code in the space of n × n symmetric matrices over Fq is a maximum symmetric rank-distance code, particularly for n > 3.

Background

A complete symmetric rank-distance code is maximal under inclusion among codes with a fixed minimum distance, whereas a maximum symmetric rank-distance code attains the relevant Singleton-type dimension bound. The paper proves that these notions coincide for the cases considered with odd q and identifies counterexamples when q is even for n=3 and d=2. It explicitly leaves open the general characterization of when the converse implication holds or fails, especially in dimensions greater than three.

References

While every MSRD code is a CSRD code, obtaining the conditions under which the converse fails remains an open question, particularly for n > 3.

— Linear complete symmetric rank-distance codes  (2503.02586 - Alnajjarine et al., 4 Mar 2025) in Section 6, “Final Remarks”