Duplicated Steiner triple systems with self-orthogonal near resolutions

TL;DR

This paper introduces methods for constructing self-orthogonal near resolvable duplicated Steiner triple systems, resolving their existence for almost all sizes and addressing a recent open question in combinatorial design theory.

Contribution

It provides new construction techniques for self-orthogonal near resolvable DSTS and determines their existence for nearly all parameter values, with only four exceptions.

Findings

Constructed self-orthogonal near resolvable DSTS for almost all v

Resolved an open question by Bryant, Davies, and Neubecker

Established existence results with only four possible exceptions

Abstract

A Steiner triple system, STS$(v)$, is a family of $3$-subsets (blocks) of a set of $v$ elements such that any two elements occur together in precisely one block. A collection of triples consisting of two copies of each block of an STS is called a duplicated Steiner triple system, DSTS. A resolvable (or near resolvable) DSTS is called self-orthogonal if every pair of distinct classes in the resolution has at most one block in common. We provide several methods to construct self-orthogonal near resolvable DSTS and settle the existence of such designs for all values of $v$ with only four possible exceptions. This addresses a recent question of Bryant, Davies and Neubecker.