We will construct (2n,2,1) resolvable BIBD with the help of difference systems.
We already proved that difference systems cannot create a resolvable BIBD in their original way, so we will change the method. We take v0 =v−1 = 2n−1 points and construct a (2n−1,2,1) BIBD with difference sets and later add the blocks with the 2nth point to complete the BIBD to a (2n,2,1) resolvable BIBD.
The (2n−1,2,1) BIBD hasv0 = 2n−1,k0 = 2,r0 = 2n−2 andb0 = (2n−1)·(n−1) = 2n2−3n+ 1. We need difference sets to cover all the non-zero v0 −1 = 2n−2 differences:
{0,2},{2n−2,3}, ...{n+ 1, n}
Those sets give the differences:
(2,2n−1−2),(4,2n−1−4), ...(2n−2,1)
Figure 5.2 shows the difference sets and their translates for the case of the (7,2,1) BIBD. Now we have a BIBD for v0 = 2n−1. We add the 2nth point ∞ to this BIBD. This is simply done by adding the block{∞,1} to our difference sets which, together with this block, form our initial blocks for the (2n,2,1) resolvable BIBD.
While on the other initial blocks both integers are incremented mod 2n−1, we develop in {∞,1} only the integer (how to increment ∞ anyway?). Figure 5.3 shows the initial blocks on the left followed by the translates giving all the blocks of the (8,2,1) resolvable BIBD. The remaining question is, if those BIBD are resolvable. Indeed, the initial blocks and their translates each correspond to a parallel class. Also v ≡k modk(k−1)≡2 mod 2, i.e. v even, is the case.
Figure 5.2: (7,2,1)-BIBD, on the upper left the difference sets, followed by their translates
2 1
Figure 5.3: (8,2,1)-BIBD, on the upper left the initial blocks, followed by their translates
6
Chapter 6Discussion
The goal was to find for arbitrary k and v with k|v methods and algorithms to construct designs.
To achieve the goal in finding methods to construct rounds we did an intensive literature research and found concepts which suit our needs: RBIBD and optimal KP. We discovered that they have already been proven (RBIBD:k∈ {2,3,4}, λ= 1 and optimal KP for v ≥18). We discovered that the approaches to get BIBD are not directly reusable for RBIBD. However, ideas of one method are reused in one of the explained methods: Galois fields and the method of pure and mixed differences are adapted for RBIBD. Other methods (Frost, Gill, Anstice) are mentioned, but do not give a lot of designs and are hard to be transformed into algorithms. Walecki’s method and the method of pure and mixed differences for resolvable designs cover more RBIBD and we contributed two algorithms to describe them. For KP, we took the theorem of doubling construction and contributed a method with specific practical ideas and an extension to use it. We then presented some other direct constructions for k= 3 with an overview, showing that most KTS and KP until v = 66 can be constructed by the methods described, only four need deeper research as in how to specifically construct them. We described a method to get all RBIBD with k= 2, only methods for k >3 were not found. However, it seems that Sage uses an algorithm to construct a small family of designs with k = 4.
Not that successful were our attempts in combining designs of different sizes for k = 3 and in combining two (v,2,1) to one (2v,4,1). Shuffling two designs with k = 3 together is difficult. If we try to combine any design with three points and shuffle them after the first round, the three points replace the elements of the other design and we combine them in a block. These three connections cannot be used later, thus making specific blocks in later rounds unusable. This destroys the structure of the design with holey classes. Of course, maybe there is a pattern creatable such that we combine the holey classes together in a different way. Another idea was to merge two (v,2,1) to one (2v,4,1)-RBIBD. This thought did not bring the desired designs, as there were no patterns in the blocks of the (v,2,1) after which we would have been able to group them as new blocks. Our original idea was to merge six blocks of a (v,2,1) into one block of a (2v,4,1). But without
a transferable pattern, a block of a (v,2,1) is simply a necessary connection in a (2v,4,1)-RBIBD, thus not helpful. New methods are obviously not that easily invented.
All together, we covered most of the topics related to RBIBD: PBD, BIBD, RBIBD and their characteristics: Blocks, rounds and existence. We collected and described various of their construction methods with focus on k = 2 andk = 3 and gathered methods or direct examples for thev with k = 3 and v ≤66, which are far more than Both et al. used in this overview [10]. We did the same steps for KP and collected the material in more extensive amounts and with more examples and explanations than the current literature and provided definitions for KP and NKTS, which are also not clearly defined in the literature so far. Furthermore, the notation for designs is different in every area and the field of economics names concepts completely different than mathematicians do. We contributed unification of these notations. We merged the mathematical expertise with the current economical standards in the design of experiments in a way, such that non-mathematicians can grasp the ideas and concepts of RBIBD more easily. We improved the definition and derivations for the upper bound of rounds M(v) which will help economics and others to improve their algorithms with a clearly defined and proven maximal number of roundsM(v). Both et al. [10] wrote that "Even small problem instances of the SGP [Social Golfer Problem] and, hence of PSM, are computationally expensive, due to the inability to determine if the maximum number of matches have been found." As we have proven, thisM(v) is very clear now. Then we took the theorem of doubling construction and created a method. We extended this method to obtain more optimal KP with non-perfect but in practice sufficient results. We even exemplified on two methods (Walecki and RPnM) how algorithms made of these methods look like. So we see, there are many possibilities to construct designs and we showed that most of the smaller ones for blocks of size 3 and all of block size 2 can be constructed easily.
There are still topics and areas related to our designs which need to be taken into account: The Oberwolfach problem as well as Sage with its code and theorems from Design Theory [7]. Furthermore, we have ideas in mind which might give us more families of KP: Merging two KP instead of KTS in the doubling construction or applying Walecki’s method on KP. There might be quite some methods which can be transformed and combined to construct different families of designs. The code from Sage seems to produce all KTS(v), so that is definitely worth another look.
Also, there are the methods of Kotzig and Rosa [23], which even the mathematician Pegg [37] could not reproduce anymore. All these methods and concepts are for v with k|v. However, there are non-optimal KP and also Kirkman covering designs,
which might be interesting for practical purposes as well. Those are out of scope, but might be interesting for the design of other experiments.
7
Chapter 7Conclusion
This overview provides the foundation to explore the practicality of RBIBD and optimal KP in the design of experiments in order to create designs with fixed block sizek andv points,k|v. We collected designs, examined their usefulness, discussed methods to construct them, created a method, designed algorithms to show how to implement them and compared the number of rounds of our results with the rounds created by Both et al. [10] . The found material was then sorted, compared to the latest results in the field of economics and explored for practicality.
In this thesis, the focus was on collecting and organzing the different topics regarding everything which is related to RBIBD and their role and use in the design of experiments. We clarified the mathematical restrictions and possibilities of those designs and the Kirkman packing designs, recognizing that the practical issues are easy to understand, but difficult to solve. For this, we collected methods;
some of them being important in general for BIBD, most of them being used to construct RBIBD or KP and sometimes depending on the general methods or using them in some way. We saw that for k = 2 all RBIBD can be constructed with ease,. For k = 3 quite some methods do construct a lot of them. An overview shows the results of this thesis regarding k = 3. We did transform some of the methods into algorithms, also we clarified the upper bound for arbitraryk, giving a proof and presenting the results for various k. Further, we transformed the theorem of doubling construction, introduced a method and showed the difficulties and possibilities for improvement in this method. We introduced an extension to this method to construct a family of non-optimal Kirkman packing designs with r= M(v)−1. For biggerkthis field is mostly unexplored and needs more attention to gain practically useful methods. We see that the presented methods have the potential to construct their designs a lot faster than any randomized or in-depth search.
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Appendix
Further Examples and Concepts
Pure and Mixed Differences: KTS(21)
An example for the method of pure and mixed differences is the KTS(21) (see theorem 5.8): v = 21, thus t= 21−36 = 3
The initial blocks are
[11,61,02],[21,51,02],[31,41,02],[12,62,03],[22,52,03], [32,42,03],[13,63,01],[23,53,01],[33,43,01],[01,02,03].
Their six translates for the fixed suffixes are:
[21,01,12],[31,61,12],[41,51,12],[22,02,13],[32,62,13], [42,52,13],[23,03,11],[33,63,11],[43,53,11],[11,12,13],
[31,11,22],[41,01,22],[51,61,22],[32,12,23],[42,02,23], [52,62,23],[33,13,21],[43,03,21],[53,63,21],[21,22,23],
[41,21,32],[51,11,32],[61,01,32],[42,22,33],[52,12,33], [62,02,33],[43,23,31],[53,13,31],[63,03,31],[31,32,33],
[51,31,42],[61,21,42],[01,11,42],[52,32,43],[62,22,43], [02,12,43],[53,33,41],[63,23,41],[03,13,41],[41,42,43],
[61,41,52],[01,31,52],[11,21,52],[62,42,53],[02,32,53], [12,22,53],[63,43,51],[03,33,51],[13,23,51],[51,52,53],
[01,51,62],[11,41,62],[21,31,62],[02,52,63],[12,42,63], [22,32,63],[03,53,61],[13,43,61],[23,33,61],[61,62,63].
These are 70 blocks. A check (b = λ(v−1)k−1 = 21·203·2 = 70) confirms that this is the amount of blocks needed.
Unfortunately, if we try to partition the blocks into parallel classes, we will fail.
Latin Squares and MOLS
The following concepts are necessary for the theoretical background and proofs of BIBD and KP.
A.1 Definition A Latin Square on n symbols (or: of order n) is an n x n array such that each symbol occurs exactly once in each row and column.
A.2 DefinitionTwo Latin Squares A and B on n symbols are called orthogonal, if the ordered pairs (aij, bij) are all different.
A.3 Example
Take two Latin Squares of order 3 A =
Superposing them, we get the ordered pairs
possible ordered pairs, so there can be no third Latin Square orthogonal to both A and B. Hence,N(3) = 2 (see definition A.4).
A.4 Definition If the Latin Squares A1, A2, ..., Ak are such that Ai is orthogonal to Aj whenever i 6= j, we call it a set of k mutually orthogonal Latin Squares (MOLS). Special interest is paid inN(n), which we define as the biggest k for a set
of order n, i.e. the largest possible set of MOLS of order n.
A and B from example A.3 form a set of 2 MOLS.