Upper bounds on the length of quasi-MDS codes

2026-07-29

AI Summary

This study investigates the maximum possible length of linear quasi-MDS codes over finite fields under the folded Hamming metric, with a focus on characterizing its dependence on the field size and other parameters. By establishing a correspondence between quasi-MDS codes and families of subspaces, the problem of bounding code length is transformed into a 1-subspace packing problem, which is further reduced to a partial spread problem in finite geometry. This work pioneers the integration of partial spread theory into the analysis of quasi-MDS code lengths, thereby enabling a novel cross-disciplinary methodological transfer. Leveraging classical bounds by Drake–Freeman, Năstase–Sissokho, and Honold–Kiermaier–Kurz, the authors not only recover the Griesmer-type upper bound previously established by Ball et al., but also derive strictly tighter length bounds across several parameter regimes.

Abstract

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, Năstase-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.

Problem

• quasi-MDS codes
• length upper bounds
• folded Hamming distance
• finite field size
• subspace packings

Innovation

• quasi-MDS codes
• subspace packings
• partial spreads
• finite geometry
• upper bounds