Large Sets of t-Designs

Citation

Ajoodani-Namini, Shahin (1998) Large Sets of t-Designs. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/y84m-hs59. https://resolver.caltech.edu/CaltechTHESIS:11202019-172054366

Abstract

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.

Thesis Files