Improved upper bounds for partial spreads

A partial $(k-1)$-spread in $\operatorname{PG}(n-1,q)$ is a collection of $(k-1)$-dimensional subspaces with trivial intersection, i.e., each point is covered at most once. So far the maximum size of a partial $(k-1)$-spread in $\operatorname{PG}(n-1,q)$ was known for the cases $n\equiv 0\pmod k$, $n\equiv 1\pmod k$ and $n\equiv 2\pmod k$ with the additional requirements $q=2$ and $k=3$. We completely resolve the case $n\equiv 2\pmod k$ for the binary case $q=2$.

💡 Research Summary

The paper investigates the maximum size of a partial ((k-1))-spread in the projective space (\operatorname{PG}(n-1,q)). A partial ((k-1))-spread is a collection of ((k-1))-dimensional subspaces that intersect trivially, i.e., each point of the space is covered by at most one subspace. In coding‑theoretic language, the size of a maximum partial ((k-1))-spread equals the constant‑dimension subspace code parameter (A_q(n,2k;k)), the largest possible cardinality of a code in (G_q(n,k)) with minimum subspace distance (2k).

Background.
If (k) divides (n), a full (k)-spread exists and the obvious bound \