New lower bounds for partial k‐parallelisms

Due to the applications in network coding, subspace codes and designs have received many attentions. Suppose that $k\mid n$ and $V\left(n,q\right)$ is an $n$‐dimensional space over the finite field ${\mathbb{F}}_q$. A $k$‐spread is a $(q^n?1)/(q^k?1)$‐set of $k$‐dimensional subspaces of $V\left(n,q\right)$ such that each nonzero vector is contained in exactly one element of it. A partial $k$‐parallelism in $V\left(n,q\right)$ is a set of pairwise disjoint $k$‐spreads. As the number of $k$‐dimensional subspaces in $V\left(n,q\right)$ is ${\left.\genfrac{}{}{0ex}{}{n}{k}\right]}_q$, there are at most ${\left.\genfrac{}{}{0ex}{}{n?1}{k?1}\right]}_q$ spreads in a partial $k$‐parallelism. By studying the independence numbers of Cayley graphs associated to a special type of partial $k$‐parallelisms in $V\left(n,q\right)$, we obtain new lower bounds for partial $k$‐parallelisms. In particular, we show that there exist at least $\frac{q^k?1}{q^n?1}{\left.\genfrac{}{}{0ex}{}{n?1}{k?1}\right]}_q$ pairwise disjoint $k$‐spreads in $V\left(n,q\right)$.