Triangle decompositions of PG(n-1,2)

TL;DR

This paper introduces balanced triangle decompositions of projective spaces over GF(2), constructs infinite classes of such designs, and explores their symmetry properties related to Singer cycles.

Contribution

It constructs balanced triangle designs in PG(n-1,2) for all n ≡ 1 mod 6 and establishes their equivalence to certain partitions of cyclic groups, advancing the understanding of subspace designs.

Findings

Constructed balanced triangle designs for all admissible n

Established equivalence with partitions of cyclic groups

Found explicit partitions for n=7, 13, 19

Abstract

We define a triangle design as a partition of the set of lines of a projective space into triangles, where a triangle consists of three pairwise intersecting lines with no common point. A triangle design is balanced if all points are involved in the same number of triangles. We construct balanced triangle designs in PG$(n-1,2)$ for all admissible $n$ (congruent to $1$ modulo $6$) and an infinite class of balanced block-divisible triangle designs. We also prove that the existence of a triangle design in PG$(n-1,2)$ invariant under the action of the Singer cycle group is equivalent to the existence of a partition of $Z_{2^n-1}\backslash\{0\}$ into special $18$-subsets and find such partitions for $n=7$, $13$, $19$. Keywords: Subspace design, graph decomposition, triangle design, Heffter's difference problem.