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arXiv · 1204.2589

1-Overlap Cycles for Steiner Triple Systems

Victoria Horan·Glenn Hurlbert
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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.

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Victoria Horan, Glenn Hurlbert. 2012-04-11. 1-Overlap Cycles for Steiner Triple Systems. https://arxiv.org/abs/1204.2589

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