Thesis (M.S.)--Virginia Polytechnic Institute, 1955. / Vita. Abstract. Includes bibliographical references (leaf 38). Also available via the Internet.

When defining a structure to fulfill a set of axioms that are similar to those prescribed by Euclid, one must select a set of points and then define what is meant by a line and what is meant by a circle. When properly defined these labels will have properties which are similar to their counterparts in the (complex) plane, the lines and circles which Euclid undoubtedly had in mind. In this manner, the geometer may employ his intuition from the complex plane to prove theorems about other systems. Most "finite geometries" have clearly defined notions of points and lines but fail to define circles. The two notable exceptions are the circles in a finite affine plane and the circles in a Mobius plane. Using the geometry of Euclid as motivation, we strive to develop structures with both lines and circles. The only successful example other than the complex plane is the affine plane over a finite field, where all of Euclid's geometry holds except for any assertions involving order or continuity. To complement the prolific work concerning finite geometries and their lines, we provide a general definition of a circle, or more correctly, of a collection of circles and present some preliminary results concerning the construction of such structures. Our definition includes the circles of an affine plane over a finite field and the circles in a Mobius plane as special cases. We develop a necessary and sufficient condition for circularity, present computational techniques for determining circularity and give varying constructions. We devote a chapter to the use of circular designs in coding theory. It is proven that these structures are not useful in the theory of error-correcting codes, since more efficient codes are known, for example the Reed-Muller codes. However, the theory developed in the earlier chapters does have applications to Cryptology. We present five encryption methods utilizing circular structures.

Balanced Incomplete Block Designs and Binary Linear Codes are two combinatorial designs. Due to the vast application of codes in communication the field of coding theory progressed more rapidly than many other fields of combinatorial designs. On the other hand, Block Designs are applicable in statistics and designing experiments in different fields, such as biology, medicine, and agriculture. Finding the relationship between instances of these two designs can be useful in constructing instances of one from the other. Applying the properties of codes to corresponding instances of Balanced Incomplete Block Designs has been used previously to show the non-existence of some designs. In this research the relationship between (16,6,3)-designs and (25,12) codes was determined.

By a study of the duality relationships of a large number of balanced and partially balanced incomplete block designs, certain ones have been found which lend themselves nicely to interblock analysis. Besides facilitating this analysis, these designs make possible the use of a new method for studying the relative variability of the inter and intra-block error. These "nice" designs, which are called twice balanced, have the property that their duals are also balanced or partially balanced. In the partially balanced designs, the investigation has been confined to those with two associate classes. Some methods are shown which may be used to prove that a dual is twice balanced. The twice balanced designs which have been found are catalogued, showing the plan numbers of the design and the dual,and the necessary identifying parameters or both. The proofs used in verifying the designs to be twice balanced are also indicated. Finally, there is an illustrative example making use of the methods and tables introduced in this paper. It includes a new computing method to be used for finding estimates of the treatment effects in a mixed model experiment. / Master of Science

We study a family of random walks defined on certain Euclidean lattices that are related to incidence matrices of balanced incomplete block designs. We estimate the return probability of these random walks and use it to determine the asymptotics of the number of balanced incomplete block design matrices. We also consider the problem of collisions of independent simple random walks on graphs. We prove some new results in the collision problem, improve some existing ones, and provide counterexamples to illustrate the complexity of the problem.

The purpose of this paper is to present a new nonparametric test statistic for testing against ordered alternatives in a Balanced Incomplete Block Design (BIBD). This test will then be compared with the Durbin test which tests for differences between treatments in a BIBD but without regard to order. For the comparison, Monte Carlo simulations were used to generate the BIBD. Random samples were simulated from: Normal Distribution; Exponential Distribution; T distribution with three degrees of freedom. The number of treatments considered was three, four and five with all the possible combinations necessary for a BIBD. Small sample sizes were 20 or less and large sample sizes were 30 or more. The powers and alpha values were then estimated after 10,000 repetitions.The results of the study show that the new test proposed is more powerful than the Durbin test. Regardless of the distribution, sample size or number of treatments, the new test tended to have higher powers than the Durbin test.

M.S.

We know that the best linear combination of the intra- and inter-block estimates is Intra-estimatexInter-variance+Inter-estimatexIntra-variance / Intra-variance + Inter-variance ; however, this combined estimate is merely theoretical, since we do not know in practice the exact inter- and intra-variances. A reasonable solution is to use a random weight which can be computed from the data of our experiment, but so far there has been no practical solution without severe restrictions on the size of the experiment, and no solution at all for a clear answer to the question of how much we recovered. In fact, the experimenter applying the methodology available to him now, cannot be sure that he is really improving the accuracy of his estimation. This research has achieved the following: 1. A new method of combining two independent estimates has been developed. This method has its use in incomplete block designs, in similar experiments, and in randomized block designs with heterogeneous variances. The improvement introduced by this method is very satisfactory, compared with the utmost possible theoretical improvement. 2. A procedure for recovering the inter-block information in B.I.B. designs was given, which is applicable in experiments as small as t = 4. 3. It has been proven that the practical utilization of inter-block information is possible in any P.B.I.B. with seven treatments or more. 4. A general procedure for recovering the inter-block information in P.B.I.B.'s with two associate classes was given. 5. An inter-block analysis of singular and semi-regular group divisible designs was discussed, which makes a partial utilization of the inter-information possible. In general, this work has two merits: 1. It makes possible the utilization of the interblock information in small and moderate size experiments. 2. As a ratio of the utmost possible theoretical recovery (by combining linearly), either exactly or a lower bound of the ratio of recovery is always computable. Tables which enable the experimenter to use the procedures described in this dissertation were given. The ratios of recovery listed in these tables show that the new method gives good results where the old method is not applicable, and when the old method starts, hopefully, to be valid, the ratio of recovery achieved by the new method starts to approach the theoretical value that can be achieved, assuming the intra- and inter-variance are known. / Ph. D.

The work of Kramer and Bradley on the use of factorials in incomplete block designs bas been extended to permit both the intra-block and combined intra- and inter-block analyses of factorials in balanced and partially balanced incomplete block designs. In particular, we have obtained a combined intra- and inter-block analysis for factorials in balanced incomplete block designs, group divisible designs, and Latin Square types of partially balanced, incomplete block designs. For the class of Latin Square sub-type L₃ designs both the intra-block and combined intra- and inter-block analyses have been developed. In general, factorial treatment combinations were assigned to the association schemes by permitting the rows of the association schemes to represent the levels of one factor and the columns to represent the levels of a second factor. The extension to multifactor factorials was then carried out by sub-dividing the levels of the basic two-factor factorial, the levels in the sub-divisions representing the levels of the multi-factor factorials. Estimators for the factorial effects have been obtained along with their variances and covariances. Sums of squares in terms of the factorial estimators have been derived and can be used to carry out tests of significance. These sums of squares were shown, for the combined intra- and inter-block analyses, to be independently distributed as χ²-variates with the appropriate numbers of degrees of freedom. Suitable sums of squares for tests of significance are not possible in general for Latin Square sub-type L₃ designs. In situations such as these, we can only consider contrasts among the estimators and use their variances to perform tests of significance. However, for the special cases of factorials in the 4 x 4 Latin Square sub-type L₃ design, a complete analysis yielding the adjusted sums of squares for the factorial effects is possible if the factorial treatments are applied to the association scheme in a different manner. Single-degree-of-freedom contrasts are obtained in much the usual way as for factorials in complete block designs. The method of incorporating a fractional replicate of a factorial into incomplete block designs is also considered. Numerical examples have been worked in detail for a group divisible design and a Latin Square sub-type L₂ design. The procedure for analyzing a Latin Square sub-type L₃ design is also discussed. / Ph. D.

Cyclic designs are incomplete block designs consisting of sets of blocks where in each set successive blocks are obtained in a cyclic manner from the first block of the set. It is convenient to label the objects (treatments) by the integers 0, 1, …, n-. Then if for a particular set the first block consists of, say, the objects 0, h, i, j, then the second block would contain 1, h+1, i+1, j+1. The remaining blocks in the set are obtained similarly, with reduction modulo n whenever an object label exceeds n-1. A cyclic design consists of such sets or combinations of sets, and will be said to be of size (n, k, r) if the block size is k and the object occurs r times. Cyclic designs are a subclass of partially balanced incomplete block (PBIS) designs. In this dissertation all non-isomorphic designs for combinations of n, k, and r are enumerated, and their efficiencies computed. Non-isomorphic designs are those which are not derivable from any other by a relabeling or permutation of the object labels. A method of analysis is presented which utilizes the cyclic property of the designs. The efficiencies are compared with those of two-associate class PBIB designs and a discussion on the utility of the designs is also given. / Ph. D.