论文标题
Motzkin号码和标志代码
Motzkin numbers and flag codes
论文作者
论文摘要
Motzkin数量已经广泛研究，因为它们计算了许多不同的组合对象。在本文中，我们通过称为标志代码的特定案例在网络编码设置中介绍了此非凡序列的新外观。标志代码是在有限字段$ \ mathbb {f} _q $上的矢量空间的嵌套子空间（标志）序列集。如果尺寸列表为$（1，\ dots，n-1）$，我们会谈论完整的标志代码。标志距离定义为相应的子空间距离的总和，可以通过所谓的距离向量表示。我们表明，$ \ mathbb {f} _q^n $上与完整标志的距离相对应的距离向量的数量正是$ n $ -th motzkin编号。此外，我们可以确定整数序列，该整数序列计算与规定最小距离的完整标志代码相关的可能距离向量的数量。
Motzkin numbers have been widely studied since they count many different combinatorial objects. In this paper we present a new appearance of this remarkable sequence in the network coding setting through a particular case of multishot codes called flag codes. A flag code is a set of sequences of nested subspaces (flags) of a vector space over the finite field $\mathbb{F}_q$. If the list of dimensions is $(1, \dots, n-1)$, we speak about a full flag code. The flag distance is defined as the sum of the respective subspace distances and can be represented by means of the so-called distance vectors. We show that the number of distance vectors corresponding to the full flag variety on $\mathbb{F}_q^n$ is exactly the $n$-th Motzkin number. Moreover, we can identify the integer sequence that counts the number of possible distance vectors associated to a full flag code with prescribed minimum distance.
