Universal Cycles on Affine Lines

TL;DR

This paper proves the existence of universal cycles on affine lines in affine geometries over finite fields, using a novel embedding into projective spaces and recursive construction methods.

Contribution

It introduces a new construction method for universal cycles on affine lines in affine geometries, extending to Grassmannians and providing a Python implementation.

Findings

Universal cycles exist for affine lines in AG(n,q) for all n ≥ 2 and prime powers q.

Construction uses embedding into PG(n,q) with points at infinity to encode directions.

The approach extends known Grassmannian constructions and includes a Python implementation.

Abstract

A universal cycle is a cyclic sequence in which each object of a combinatorial family appears exactly once as a contiguous window. While such cycles are well understood for many discrete structures and linear subspaces, the case of affine lines presents additional difficulties arising from parallelism. We prove that universal cycles exist for affine lines in $\mathrm{AG}(n,q)$ for all $n \ge 2$ and all prime powers $q$. Our construction embeds the problem into $\mathrm{PG}(n,q)$, using points at infinity to encode directions, and proceeds via a decomposition into pairwise and triple configurations combined with a recursive lifting and gluing argument. We further interpret the construction in the Grassmannian $G_q(2,n+1)$, where affine lines correspond to the outer shell of $2$-subspaces, thereby extending known constructions for Grassmannians. A Python implementation is provided as supplementary material.