Grassmannian $${{{\mathcal {G}}}}_q(n,k)$$ is the set of all k-dimensional subspaces of the vector space $${\mathbb {F}}_q^n$$. Kotter and Kschischang showed that codes in Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are q-analogs of codes in the Johnson scheme, i.e. constant dimension codes. These codes of the Grassmannian $${{{\mathcal {G}}}}_q(n,k)$$ also form a family of q-analogs of block designs and they are called subspace designs. In this paper, we examine one of the last families of q-analogs of block designs which was not considered before. This family called subspace packings is the q-analog of packings, and was considered recently for network coding solution for a family of multicast networks called the generalized combination networks. A subspace packingt-$$(n,k,\lambda )_q$$ is a set $${\mathbb {S}}$$ of k-subspaces from $${{{\mathcal {G}}}}_q(n,k)$$ such that each t-subspace of $${{{\mathcal {G}}}}_q(n,t)$$ is contained in at most $$\lambda $$ elements of $${\mathbb {S}}$$. The goal of this work is to consider the largest size of such subspace packings. We derive a sequence of lower and upper bounds on the maximum size of such packings, analyse these bounds, and identify the important problems for further research in this area.

中文翻译：

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子空间包装：构造和边界

Grassmannian $${{{\mathcal {G}}}}_q(n,k)$$ 是向量空间 $${\mathbb {F}}_q^n$$ 的所有 k 维子空间的集合。Kotter 和 Kschischang 表明，格拉斯曼空间中的代码可用于随机网络编码中的纠错。另一方面，这些代码是约翰逊方案中代码的 q 类比，即常数维代码。Grassmannian $${{{\mathcal {G}}}}_q(n,k)$$ 的这些代码也形成了一个块设计的 q 模拟族，它们被称为子空间设计。在本文中，我们研究了以前未考虑过的最后一个模块设计 q 模拟系列之一。这个称为子空间打包的族是打包的 q 类比，最近被认为是用于称为广义组合网络的多播网络族的网络编码解决方案。一个子空间打包t-$$(n,k, \lambda )_q$$ 是来自 $${{{\mathcal {G}}}}_q(n,k)$$ 的 k 子空间的集合 $${\mathbb {S}}$$，使得每个 t -$${{{\mathcal {G}}}}_q(n,t)$$ 的子空间最多包含在 $${\mathbb {S}}$$ 的 $$\lambda $$ 元素中。这项工作的目标是考虑这种子空间包装的最大尺寸。我们推导出此类填料最大尺寸的下限和上限序列，分析这些界限，并确定该领域进一步研究的重要问题。