Upper bounds on the length of quasi-MDS codes

We study upper bounds on the length of $\mathbb F_q$-linear QMDS codes in the folded Hamming distance relative to their other parameters, especially the field size $q$. Via a correspondence between such codes and families of subspaces, we relate the length problem to that of upper bounding $1$-subspace packings with respect to the other parameters, especially the field size. Our main result is a reduction from these families to partial spreads, which allows us to import sharp bounds from finite geometry, including results of Drake-Freeman, N\u{a}stase-Sissokho, and Honold-Kiermaier-Kurz. As a consequence, we recover the Griesmer-type upper bound on the length of QMDS codes by Ball et al. and obtain tighter upper bounds in several parameter regimes.