New lower bounds for partial $k$-parallelisms

Abstract

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 ${\left[\genfrac{}{}{0ex}{}{n}{k}\right]}_q$, 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 $V(n,q)$, we obtain new lower bounds for partial $k$-parallelisms. In particular, we show that there exist at least $\frac{q^k-1}{q^n-1}{\left[\genfrac{}{}{0ex}{}{n-1}{k-1}\right]}_q$ pairwise disjoint $k$-spreads in $V(n,q)$.

To appear in Journal of Combinatorial Designs

References in corpus (1)

• Automorphisms of Codes in the Grassmann Scheme