New lower bounds for partial $k$-parallelisms

TL;DR

This paper establishes new lower bounds for the size of partial $k$-parallelisms in finite vector spaces, leveraging Cayley graph independence numbers, with implications for network coding and combinatorial designs.

Contribution

It introduces novel lower bounds for partial $k$-parallelisms by analyzing Cayley graph independence numbers, advancing understanding of subspace arrangements.

Findings

Existence of at least rac{q^{k}-1}{q^{n}-1}{n-1 rack k-1}_q disjoint $k$-spreads.

New lower bounds improve previous estimates for partial $k$-parallelisms.

Application of Cayley graph analysis to combinatorial design problems.

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 ${n \brack k}_{q}$, 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 $V(n,q)$.