Technical report Existence of Kirkman signal sets on $v=1,3\pmod{6}$ points, $14\leq v \leq 3000$

TL;DR

This paper investigates the existence of Kirkman signal sets, a special type of partial Steiner triple systems, for certain values of v, providing a table of known existence results within the range 14 to 3000.

Contribution

It establishes the existence conditions for Kirkman signal sets on points where v ≡ 1 or 3 mod 6, and compiles a comprehensive table of known results for these parameters.

Findings

Existence results for Kirkman signal sets on specified v values.

Complete table of known existence results for 14 ≤ v ≤ 3000.

Characterization of the relationship between partial Steiner triple systems and Kirkman signal sets.

Abstract

A partial Steiner triple system whose triples can be partitioned into $s$ partial parallel classes, each of size $m$, is a $signal$ $set$, denoted $\mbox{SS}(v,s,m)$. A $Kirkman$ $signal$ $set$ $\mbox{KSS}(v,m)$ is an $\mbox{SS}(v,s,m)$ with $s=\lfloor\mu(v)/m\rfloor$. When $v \equiv 1$ or $3 \pmod{6}$, then $\mu(v)=b$, so the decomposition of an $\mbox{STS}(v)$ into partial parallel classes of size $m$ is equivalent to a $\mbox{KSS}(v,m)$. Table of known existence results is given.