3,572 research outputs found
On large sets of disjoint steiner triple systems III
AbstractTo construct large sets of disjoint STS(3n) (i.e., LTS(3n)), we introduce a new kind of combinatorial designs. Let S be a set of n elements. If x ∈ S, we denote an n × n square array on S by Ax, if for every w ∈ S\{x} the following conditions are satisfied: Ax = [ayz(x)](y, z ∈ S), axx(x) = x, aww(x) ≠w, axw(x) = axw(x) = x, and {awz(x) | z ∈ S} = {ayw(x) | y ∈ S} = S. Let j ∈ {1, 2&}, Aj = {ayz[j](y, z ∈ S) be a Latin square of order n based on S with n parallel transversals including the diagonal one. Two squares Ax and Ax′ on the same S are called disjoint, if ayz(x) ≠ayz(x′) whenever y, z ∈ S\{x, x′}; two squares Ax and Aj on the same S are called disjoint, if ayz(x) ≠ayz[j] whenever y, z ∈ S\ {x}; and two squares A1 and A2 on the same S are called disjoint, if ayz[1] ≠ayz[2] whenever y ≠ez. It is a set of n + 2 pairwise disjoint squares Ax (x runs over S), A1 and A2 on S as mentioned above that is very useful to construct LTS(3n), and such a set we denote by LDS(n). The essence in the relation between LDS(n) and LTS(3n) is the following theorem which is established in the Section 2:Theorem. If there exist both an LDS(n) and an LTS(n + 2), then there exists an LTS(3n) also.The set of integers n for which LDS(n) exist is denoted by D. In the other parts of this paper, the following results are given: 1.(1) If n ∈ D, and q = 2α (α is an integer greater than 1), or q ∈ {;5, 7, 11, 19}, then qn ∈ D.2.(2) If pα is a prime power, p > 2 and pα ∈ D, then 3pα ∈ D.3.(3) If q is a prime power greater than 4 and 1 + n ∈ D, then 1 + qn ∈ D.4.(4) If t is a nonnegative integer, then 7 + 12t ∈ D and 5 + 8t ∈ D
Steiner triple systems and spreading sets in projective spaces
We address several extremal problems concerning the spreading property of
point sets of Steiner triple systems. This property is closely related to the
structure of subsystems, as a set is spreading if and only if there is no
proper subsystem which contains it. We give sharp upper bounds on the size of a
minimal spreading set in a Steiner triple system and show that if all the
minimal spreading sets are large then the examined triple system must be a
projective space. We also show that the size of a minimal spreading set is not
an invariant of a Steiner triple system
Symmetric quasigroups of odd order
AbstractQuasigroups of yet another type turn out to be related to Steiner Triple Systems, though the connection is rather loose and not as precise as in the various coordinatizing bijections described in [3]. However, families of pairs formed by abelian groups of odd order and quasigroups defined on the same set of elements have repeatedly been used in the literature to construct Large Sets [8] of Steiner Triple Systems. In Section 1, these quasigroups and their association with abelian groups are described, while Section 2 is devoted to applications to STSs
An explicit construction for large sets of infinite dimensional -Steiner systems
Let be a vector space over the finite field . A
-Steiner system, or an , is a collection of
-dimensional subspaces of such that every -dimensional subspace of
is contained in a unique element of . A large set of
-Steiner systems, or an , is a partition of the -dimensional
subspaces of into systems. In the case that has infinite
dimension, the existence of an for all finite with
was shown by Cameron in 1995. This paper provides an explicit construction of
an for all prime powers , all positive integers , and
where has countably infinite dimension.Comment: 5 page
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Combinatorial designs and their automorphism groups
This thesis concerns the automorphism groups of Steiner triple systems and of cycle systems. Although most Steiner triple systems have trivial automorphism groups [2], it is widely known that for every abstract group, there exists a Steiner triple system whose automorphism is isomorphic to that group [16]. The well-known Bose construction [4] for Steiner triple systems, which has a number of variants, has a particularly nice structure, which makes it possible to say much about the automorphism group, and in the case of the construction based on an Abelian group, to derive the full automorphism group. The thesis contains a full analysis of these matters. Some of these results have been published by the author in [14]. The thesis also proves new results concerning the automorphism group for Steiner triple systems constructed using the tripling construction. An m-cycle system is a decomposition of a complete graph into cycles of length m. A Steiner triple system is thus a 3-cycle system. The thesis proves the result that for all m > 3, and for each abstract finite group, there exists an m-cycle system whose automorphism group is isomorphic to that group. In addition, the thesis contains a collection of new results concerning the conjecture by Furedi that every Steiner triple system is decomposable into triangles. Although this conjecture is expected to remain open for some time, it is possible to prove it for a number of standard constructions. It is further shown that for sufficiently large v, the number of Steiner triple systems of order v that are decomposable into triangles is at least vv2(1/54-0(1))
Probabilistic Existence of Large Sets of Designs
A new probabilistic technique for establishing the existence of certain
regular combinatorial structures has been recentlyintroduced by Kuperberg,
Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under
certain conditions, a randomly chosen structure has the required properties of
a - combinatorial design with tiny, yet positive,
probability.
The proof method of KLP is adapted to show the existence of large sets of
designs and similar combinatorial structures as follows. We modify the random
choice and the analysis to show that, under the same conditions, not only does
a - design exist but, in fact, with positive probability
there exists a large set of such designs -- that is, a partition of the set of
-subsets of into -designs - designs.
Specifically, using the probabilistic approach derived herein, we prove that
for all sufficiently large , large sets of - designs exist
whenever and the necessary divisibility conditions are satisfied.
This resolves the existence conjecture for large sets of designs for all .Comment: 20 page
Tremain equiangular tight frames
Equiangular tight frames provide optimal packings of lines through the
origin. We combine Steiner triple systems with Hadamard matrices to produce a
new infinite family of equiangular tight frames. This in turn leads to new
constructions of strongly regular graphs and distance-regular antipodal covers
of the complete graph.Comment: 11 page
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