Universal Cycles for Affine Planes and 3-Subspaces over Finite Fields

We construct universal cycles for affine planes in ${\mathbb{F}}_q^n$ for all prime powers $q$ and all $n\ge 4$, using sliding windows of length three. The construction is local-to-global: explicit local cycles are built on frame configurations, the linear $2$-subspaces are organized by a layered frame decomposition, and the resulting cycles are assembled by gluing along shared directions. The universal cycle obtained has direction set containing all $1$-subspaces. We also extend the construction to universal cycles for $3$-subspaces of ${\mathbb{F}}_q^n$.