Diagonally-cyclic Steiner Latin squares

A Steiner triple system is a decomposition of Kn into K3, such as S={013,026,045,124,156,235,346}. Steiner triple systems give rise to a Steiner Latin squares, such as L below. L=(0361542314026564251301053624521640346320512504316) We define L=(lij) by lii=i for all i and lij=k whenever ijk is a triangle in S. Note: Typically, Steiner Latin squares are viewed in an … Read more

Solving a Diophantine equation related to Algebraic Geometry, Steiner systems and qq-binomials?

The short version of my question is: 1)For which positive integers k,n is there a solution to the equation k(6k+1)=1+q+q2+⋯+qn with q a prime power? 2) For which positive integers k,n is there a solution to the equation (3k+1)(2k+1)=1+q+q2+⋯+qn with q a prime power? Now for some motivation. In this question I ask for an … Read more

Enumerating subsets with no triple appearing together more than once

This question is motivated by a real-world application related to an art project that involves displaying images, but my search hit a dead end after finding the wikipage about Kirkman systems (other related terms include Steiner systems and the Social golfer problem) and looking over references linked there. A few people have written programs for … Read more

covering designs of the form (v,k,2)(v,k,2)

A covering design (v,k,t) is a family of subsets of [v] each having k elements such that given any subset of [v] of t elements it is a subset of one of the sets of the family. A problem is to find the minimum number of subsets such a family can have. I am interested … Read more

Isomorphism testing in STS(13)

What is the simplest isomorphism invariant which can distinguish between the two non-isomorphic Steiner triple systems on 13 points? Train structure and cycle structure, as described here, do the job, but is there a simple way? Maybe via p-ranks or something like that? Answer Take the 26-vertex graphs whose vertices are the blocks and where … Read more

Sections of “forgetful” projections between flag manifolds

Given a subset S⊆{1,⋯,n} there is an associated flag manifold F(S). Whenever A⊆B there is a “forgetful” projection F(A)←F(B) (in fact I think its fibers are direct products of flag manifolds). Due to the following examples, I am curious when these projectios have sections: From complex vector spaces, we get sections F(1,2n)→F(1,2,2n). From quaternionic vector … Read more

Database of Steiner triple systems

Can anyone point me to an online database of Steiner triple systems? My Google-fu is only getting me to descriptions of the few smallest ones, mostly Google book scans (which are rather useless to process using a computer…) Answer The first answer is not a database of the Steiner triple systems, but rather how many … Read more

Constructing Steiner Triple Systems Algorithmically

I want to create STS(n) algorithmically. I know there are STS(n)s for n≅1,3mod6. But it is difficult to actually construct the triples. For STS(7) it is pretty easy and but for larger n I end up using trial and error. Is there a general algorithm that can be used? Answer The following is Bose’s construction … Read more