近均匀 $q$-拟阵

中文摘要 AI 辅助

我们研究了与非退化 $\F_{q^m}$-线性近MRD码相关联的 $q$-拟阵，称之为近均匀 $q$-拟阵。利用秩度量码与其关联的 $q$-系统之间的对应关系，我们显式地确定了它们的秩函数并刻画了其循环平展。我们证明近均匀 $q$-拟阵构成一类可表示的铺砌 $q$-拟阵，并证明了其循环平展数量的上界，该上界对某些参数是紧的。最后，当 $n>m$ 时，利用近MRD码的秩分布，我们获得了关联 $q$-拟阵的非平凡循环平展的精确计数。

英文摘要

We study the $q$-matroids associated with nondegenerate $\F_{q^m}$-linear near-MRD codes, which we call near-uniform $q$-matroids. Using the correspondence between rank-metric codes and their associated $q$-systems, we determine explicitly their rank functions and characterize their cyclic flats. We show that near-uniform $q$-matroids form a class of representable paving $q$-matroids. and prove an upper bound on the number of their cyclic flats, which is shown to be sharp for certain parameters. Finally, when $n>m$, exploiting the rank distribution of near-MRD codes, we obtain an exact count of the nontrivial cyclic flats of the associated $q$-matroids.

发表机构

• University of Naples Federico II（那不勒斯费德里科二世大学）
• Beijing Jiaotong University（北京交通大学）

机构由 AI 辅助整理，请以论文原文为准。