1-Overlap Cycles for Steiner Triple Systems

TL;DR

This paper proves the existence of 1-overlap cycles for automorphism free Steiner triple systems of all orders, enabling compressed universal cycles and advancing combinatorial design theory.

Contribution

It establishes the existence of 1-overlap cycles for all automorphism free Steiner triple systems, extending the concept of universal cycles in combinatorial designs.

Findings

Existence of 1-overlap cycles for all orders of automorphism free Steiner triple systems.

Construction of compressed universal cycles from these 1-overlap cycles.

Extension of Dewar's results on universal cycles in Steiner triple systems.

Abstract

A number of applications of Steiner triple systems (e.g. disk erasure codes) exist that require a special ordering of its blocks. Universal cycles, introduced by Chung, Diaconis, and Graham in 1992, and Gray codes are examples of listing elements of a combinatorial family in a specific manner, and Godbole invented the following generalization of these in 2010. 1-overlap cycles require a set of strings to be ordered so that the last letter of one string is the first letter of the next. In this paper, we prove the existence of 1-overlap cycles for automorphism free Steiner triple systems of each possible order. Since Steiner triple systems have the property that each block can be represented uniquely by a pair of points, these 1-overlap cycles can be compressed by omitting non-overlap points to produce rank two universal cycles on such designs, expanding on the results of Dewar.