准MDS码长度的上界
Upper bounds on the length of quasi-MDS codes
AI总结：
该研究通过建立准MDS码与子空间族的对应关系，将其长度上界问题归约为部分弥散，借助有限几何结果得到了更紧的QMDS码长度上界。
AI中文摘要：
我们研究了相对于其他参数（尤其是域大小q）的$\boldsymbol{\text{F}}_q$-线性QMDS码在折叠汉明距离下的长度上界。通过这类码与子空间族之间的对应关系，我们将长度问题与关于其他参数（尤其是域大小）的1-子空间填充的上界问题关联起来。我们的主要结果是将这些子空间族归约为部分弥散，这使得我们能够引入有限几何中的精确上界，包括Drake-Freeman、Năstase-Sissokho以及Honold-Kiermaier-Kurz的结果。由此，我们重新得到了Ball等人提出的QMDS码长度的Griesmer型上界，并在若干参数区域获得了更紧的上界。
英文摘要：
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.
