Symmetric triangle designs and group divisible triangle designs in PG(n-1,2)
Description
This dataset is an attachment to the paper [M.Shi, X.Li, D.S.Krotov, Triangle decompositions of PG(n-1,2), Discrete Mathematics, 2025+]. Let GF(2^{19}) = GF(2)[x] / <x^{19}+x^5+x^2+x+1>; let z be a primitive root. The file triangle19.txt contains a list of 1533 pairs (i,j). Each pair represents a triangle { <1,z^i>, <1,z^{-j}>, <z^i,z^{-j}> }. The orbits of the given representatives under the action of the semidirect product of the multiplicative group of GF(2^{19}) and Aut(GF(2^{19})) form a triangle design, i.e., a partition of the set of lines of PG(19-1,2) into triangles (triples of mutually intersecting lines without a common point). The descriptions for triangle13.txt and triangle07.txt are similar, where GF(2^{13}) = GF(2)[x] / <x^{13}+x^4+x^3+x+1> and GF(2^7)=GF(2)[x]/<x^7+x+1>. The file triangle12_6.txt contains a list of 224 pairs (i,j). Each pair represents a triangle { <1,z^i>, <1,z^{-j}>, <z^i,z^{-j}> }. The orbits of the given representatives under the action of the multiplicative group of GF(2^{12}) form a group divisible triangle design, i.e., a partition of the set of lines of PG(12-1,2) that do not lie in projective 5-subspaces from the Desarguesian spread into triangles (triples of mutually intersecting lines without a common point). Here, GF(2^{12}) = GF(2)[x] / <x^{12}+x^7+x^6+x^5+x3+x+1>. The file check.sage contains a SAGE script that checks the files triangle07.txt, triangle12_6.txt, triangle13.txt, and triangle19.txt. With this script, the first three of them can be checked using the online interface at https://sagecell.sagemath.org/ . The file GD2(6,2).sage contains a SAGE script that constructs and checks a triangle group divisible design in PG(6-1,2) with Desarguesian-spread lines as groups. The design is symmetric with respect to the group 3*Sc (of order 21), where Sc in the Singer cycle group of order 63.