Universal Cycles for Minimum Coverings of Pairs by Triples, with Application to 2-Radius Sequences

TL;DR

This paper introduces a new ordering concept for set systems, proves the existence of minimal pair coverings with this ordering for all sizes, and applies it to generate short 2-radius sequences, including new sequences found computationally.

Contribution

It extends universal cycle notions to k-uniform set systems, establishing their existence for all orders and applying them to construct efficient 2-radius sequences.

Findings

Existence of such orderings for all set system sizes.

Construction of short 2-radius sequences using the new ordering.

Discovery of new 2-radius sequences via computer search.

Abstract

A new ordering, extending the notion of universal cycles of Chung {\em et al.} (1992), is proposed for the blocks of $k$-uniform set systems. Existence of minimum coverings of pairs by triples that possess such an ordering is established for all orders. Application to the construction of short 2-radius sequences is given, with some new 2-radius sequences found through computer search.