5 research outputs found
Large Sets of t-Designs
We investigate the existence of large sets of t-designs. We introduce t-wise equivalence
and (n, t)-partitionable sets. We propose a general approach to construct large
sets of t-designs. Then, we consider large sets of a prescribed size n. We partition
the set of all k-subsets of a v-set into several parts, each can be written as product
of two trivial designs. Utilizing these partitions we develop some recursive methods
to construct large sets of t-designs. Then, we direct our attention to the large sets
of prime size. We prove two extension theorems for these large sets. These theorems
are the only known recursive constructions for large sets which do not put any
additional restriction on the parameters, and work for all t and k. One of them,
has even a further advantage; it increase the strength of the large set by one, and it
can be used recursively which makes it one of a kind. Then applying this theorem
recursively, we construct large sets of t-designs for all t and some blocksizes k.
Hartman conjectured that the necessary conditions for the existence of a large
set of size two are also sufficient. We suggest a recursive approach to the Hartman
conjecture, which reduces this conjecture to the case that the blocksize is a power
of two, and the order is very small. Utilizing this approach, we prove the Hartman
conjecture for t = 2. For t = 3, we prove that this conjecture is true for infinitely
many k, and for the rest of them there are at most k/2 exceptions.
In Chapter 4 we consider the case k = t + 1. We modify the recursive methods
developed by Teirlinck, and then we construct some new infinite families of large
sets of t-designs (for all t), some of them are the smallest known large sets. We also
prove that if k = t + 1, then the Hartman conjecture is asymptotically correct.</p
Large Sets of Disjoint t-Designs
In this paper, we show how the basis reduction algorithm of Kreher and Radziszowski [4] can be used to construct large sets of disjoint designs with specified automorphisms. In particular, we construct a (3,4,23;4)-large set which gives rise to an infinite family of large sets of 4-designs via a result of Teirlinck [6]. 1 Introduction Let X be a finite set of v elements called points. We denote by \Gamma X k \Delta the set of all k-element subsets of X . A t-design, or more specifically, a t-(v; k; ) design, is a pair (X ; B) such that B ` \Gamma X k \Delta , and every member of \Gamma X t \Delta is contained in precisely members of B. The members of B are called blocks. The divisibility conditions \Gamma v\Gammai t\Gammai \Delta j 0 (mod \Gamma k\Gammai t\Gammai \Delta ) for 0 i ! t, provide necessary conditions for the existence of a t-(v; k; ) design. For any given t, k, and v, we denote by (t; k; v) the minimum positive that satisfies the divisibility ..