Even-freeness of cyclic 2-designs

TL;DR

This paper investigates the property of even-freeness in cyclic Steiner 2-designs, establishing the existence of infinitely many such designs with high even-freeness that are not derived from projective geometry.

Contribution

It proves the existence of infinitely many nontrivial cyclic Steiner 2-designs with higher even-freeness beyond the trivial bound, expanding understanding of their structure.

Findings

Existence of infinitely many nontrivial cyclic Steiner 2-designs with high even-freeness.

These designs are not derived from projective geometry.

Higher even-freeness achieved beyond the trivial lower bound.

Abstract

A Steiner 2-design of block size k is an ordered pair (V, B) of finite sets such that B is a family of k-subsets of V in which each pair of elements of V appears exactly once. A Steiner 2-design is said to be r-even-free if for every positive integer i =< r it contains no set of i elements of B in which each element of V appears exactly even times. We study the even-freeness of a Steiner 2-design when the cyclic group acts regularly on V. We prove the existence of infinitely many nontrivial Steiner 2-designs of large block size which have the cyclic automorphisms and higher even-freeness than the trivial lower bound but are not the points and lines of projective geometry.