A construction of Steiner Triple Systems of type $v\longrightarrow 2v+7$

TL;DR

The paper introduces a new construction that lifts an $STS(v)$ to an $STS(2v+7)$ in the special case $v=2^n-7$, using a difference-method embedding based on a difference-factorization of $\mathbb Z_{2^n}$ and a fixed difference triple $(1,2,3)$. By iterating this construction from a base $STS(9)$, it yields $STS(2^n-7)$ for all $n\ge 4$ with a maximal independent set of maximal cardinality and an upper chromatic number $\overline{\chi}=n-1$, i.e., an $(n-1)$-bicolorable system. This extends the classical $v\to 2v+1$ doubling construction and provides combinatorial designs with strong colorability properties relevant to design theory. The results give explicit constructive methods and bases (including an appendix example for $v=9$) to realize large $STS$ with optimized independence and colorability characteristics.

Abstract

A Steiner Triple System ($STS$) of order $v$ is a hypergraph uniform of rank 3, with $v$ vertices and such that every 2-subset of vertices has degree 1. In this paper we give a construction, by difference method, of type $v\longrightarrow 2v+7$ with $v=2^n-7$, which means that, given an $STS$ of order $v=2^n -7$, it is always possible to construct an $STS$ of order $2^{n+1}-7$. Through this construction it is possible to get for any $n\ge 5$ an $STS(2^n-7)$ with a maximal independent set of maximal cardinality and which is $(n-1)$-bicolorable.

Theorems & Definitions (16)

• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 4.1
• proof