Due to the applications in network coding, subspace codes and designs have received many attentions. Suppose that $k\mid n$ and $V(n,q)$ is an $n$-dimensional space over the finite field $\mathbb{F}_{q}$. A $k$-spread is a $\frac{q^n-1}{q^k-1}$-set of $k$-dimensional subspaces of $V(n,q)$ such that each nonzero vector is covered exactly once. A partial $k$-parallelism in $V(n,q)$ is a set of pairwise disjoint $k$-spreads. As the number of $k$-dimensional subspaces in $V(n,q)$ is ${n \brack k}_{q}$, there are at most ${n-1 \brack k-1}_{q}$ spreads in a partial $k$-parallelism. By studying the independence numbers of Cayley graphs associated to a special type of partial $k$-parallelisms in $V(n,q)$, we obtain new lower bounds for partial $k$-parallelisms. In particular, we show that there exist at least $\frac{q^{k}-1}{q^{n}-1}{n-1 \brack k-1}_q$ pairwise disjoint $k$-spreads in $V(n,q)$.

中文翻译：

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部分 k 平行的新下限

由于在网络编码中的应用，子空间编码和设计受到了很多关注。假设 $k\mid n$ 和 $V(n,q)$ 是有限域 $\mathbb{F}_{q}$ 上的 $n$ 维空间。$k$-spread 是 $\frac{q^n-1}{q^k-1}$-set 的 $k$-维的 $V(n,q)$ 子空间，使得每个非零向量是刚好覆盖一次。$V(n,q)$ 中的部分 $k$-并行是一组成对不相交的 $k$-spreads。由于$V(n,q)$中$k$维子空间的个数为${n \brack k}_{q}$，所以最多有${n-1 \brack k-1}_{ q}$ 以部分 $k$ 平行的方式展开。通过研究与 $V(n,q)$ 中特殊类型的部分 $k$-平行度相关的 Cayley 图的独立数，我们获得了部分 $k$-平行度的新下界。特别是，