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

Abstract

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 $\mathbb F_q^n$.

Disclosure

“Acknowledgment. Portions of the exposition were prepared with the as- sistance of ChatGPT. The authors also performed computer-assisted checks of the local frame constructions and gluing procedures for small prime fields and dimensions; these checks were used to detect possible indexing errors and to confirm the behavior of the”

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Classification

Drafting limited passages

Multiplier

5

Verified

2026-08-24T00:34:09.244Z

Structural counts