A quasidouble of the affine plane of order 4 and the solution of a problem on additive designs

A 2-(v,k,λ) block design (P,B) is additive if, up to isomorphism, P can be represented as a subset of a commutative group (G,+) in such a way that the k elements of each block in B sum up to zero in G. If, for some suitable G, the embedding of P in G is also such that, conversely, any zero-sum k-subset of P is a block in B, then (P,B) is said to be strongly additive. In this paper we exhibit the very first examples of additive 2-designs that are not strongly additive, thereby settling an open problem posed in 2019. Our main counterexample is a resolvable 2-(16,4,2) design (F_4×F_4, B_2), which decomposes into two disjoint isomorphic copies of the affine plane of order four. An essential part of our construction is a (cyclic) decomposition of the point-plane design of AG(4,2) into seven disjoint isomorphic copies of the affine plane of order four. This produces, in addition, a solution to Kirkman's schoolgirl problem

Computing techniques for the enumeration of cyclic Steiner systems

In this thesis a powerful algorithm is developed for finding cyclic Steiner systems. A cyclic Steiner system with parameters S(t,k,v) is a pair ( V,B), where B is a collection of subsets all of size k (called blocks) and V is a t; element set of points, such that each t-subset of V is contained in precisely one block of B. A Steiner system is called cyclic if it has an automorphism carrying the points in a v-cycle. The results obtained so far with this algorithm are given in Table VII of chapter 5. Among the values reported there, are the number of distinct cyclic solutions to S(2,3,55), S(2,3,57), S(2,3,61) and S(2,3,63) which are 121,098,240, 84,672,512, 2,542,203,904 and 1,782,918,144 respectively. These values were apparently unknown previous to this work

Infinite Jordan Permutation Groups

Abstract If G is a transitive permutation group on a set X, then G is a Jordan group if there is a partition of X into non-empty subsets Y and Z with |Z| > 1, such that the pointwise stabilizer in G of Y acts transitively on Z (plus other non-degeneracy conditions). There is a classification theorem by Adeleke and Macpherson for the infinite primitive Jordan permutation groups: such group preserves linear-like structures, or tree-like structures, or Steiner systems or a ‘limit’ of Steiner systems, or a ‘limit’ of betweenness relations or D-relations. In this thesis we build a structure M whose automorphism group is an infinite oligomorphic primitive Jordan permutation group preserving a limit of D-relations. In Chapter 2 we build a class of finite structures, each of which is essentially a finite lower semilinear order with vertices labelled by finite D-sets, with coherence conditions. These are viewed as structures in a relational language with relations L,L',S,S',Q,R. We describe possible one point extensions, and prove an amalgamation theorem. We obtain by Fra¨ıss´e’s Theorem a Fra¨ıss´e limit M. In Chapter 3, we describe in detail the structure M and its automorphism group. We show that there is an associated dense lower semilinear order, again with vertices labelled by (dense) D-sets, again with coherence conditions. By a method of building an iterated wreath product described by Cameron which is based on Hall’s wreath power, we build in Chapter 4 a group K < Aut(M) which is a Jordan group with a pre-direction as its Jordan set. Then we find, by properties of Jordan sets, that a pre-D-set is a Jordan set for Aut(M). Finally we prove that the Jordan group G = Aut(M) preserves a limit of D-relations as a main result of this thesis