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 $n\ge4$, 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 $\mathbb F_q^n$.