2012/04/11 by Victoria Horan, Horan, Victoria, Glenn Hurlbert +1
Computer Science · Engineering · #68R15 (Primary) 05B05 (Secondary) #Cellular Automata and Applications #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1204.2589
openalex publication_date 2012/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.