Cyclic, Simple and Indecomposable Three-fold Triple Systems

TL;DR

This paper constructs cyclic, simple, and indecomposable three-fold triple systems for all admissible orders using Skolem-type sequences, extending previous work on two-fold systems and employing computational verification.

Contribution

It introduces a new construction method for three-fold systems with desired properties for all admissible orders, with some exceptions, using Skolem-type sequences and computer assistance.

Findings

Successfully constructed systems for most admissible orders

Used computational methods to verify simplicity

Extended known results from two-fold to three-fold systems

Abstract

In 2000, Rees and Shalaby constructed simple indecomposable two-fold cyclic triple systems for all v congruent to 0, 1, 3, 4, 7, and 9 (mod 12) where v = 4 or v>11, using Skolem-type sequences. We construct, using Skolem-type sequences, three-fold triple systems having the properties of being cyclic, simple, and indecomposable for all admissible orders v, with some possible exceptions for v = 9 and v = 24c + 57, where c >1 is a constant. To prove the simplicity we used a Mathematica computer program. We list in the Appendix the code and the results of the program.