Boolean 2-designs and the embedding of a 2-design in a group

TL;DR

This paper investigates embedding boolean 2-designs into finite commutative groups, specifically analyzing the structure and enumeration of blocks where the sum of points is zero, with a focus on the set of non-zero vectors in $(Z_2)^n$.

Contribution

It provides a method to embed t-designs into finite groups and computes the number of blocks with zero sum in boolean 2-designs over $(Z_2)^n$.

Findings

Number of blocks with all non-zero vectors in $(Z_2)^n$

Enumeration of blocks with zero sum

Embedding conditions for t-designs in groups

Abstract

We try to embed a t-design in a finite commutative group in such a way that the sum of the k points of a block is zero. We can compute the number of blocks of the boolean 2-design having all the non zero vectors of $(Z_2)^n$ as the set of points and the k-subsets of elements the sum of which is zero as blocks.