Academic literature on the topic 'Incomplete block designs'

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.

The ever-increasing demand and requirements of communication systems necessitates research into advanced techniques to ensure reliable and optimum use of communication resources such as bandwidth. Channel characteristics such as noise, erasures and fading have a destructive effect on the transmission requiring novel approaches to obtain a reliable reconstruction at the receiver. In this thesis, we investigate error protection using linear block codes for Gaussian noise, erasure and fading channels. In the design of an error protection system, we focus on a class of near Shannon-limit approaching codes called Low-Density Parity-Check (LDPC) codes. The work presents and investigates a method of constructing LDPC codes using combinatorial mathematics known as Balanced Incomplete Block Designs (BIBD). Two classes of BIBDs are used to create new construction methods for generating quasi- cyclic (QC) LDPC matrices. The method of constructing QC-LDPC codes using BIBDs introduced offer flexible matrices while preserving the QC structure whilst ensuring efficient and low complexity encoding using linear shift registers. The structure provides similar performance to that of MacKay codes over the Gaussian channel, whilst also showing very good performance over both the random and burst erasure channels. The construction of QC-LDPC codes using BIBDs methods also result in codes with fast decoding convergence and low error floors. The introduced QC-BIBD-LDPC codes are applied to the power-line environment, which inhibits both frequency-selective fading and severe noise characteristics. The effects of the frequency-selective fading of the channel are reduced-using a Multi-carrier- Modulation (MCM) technique known as Orthogonal Frequency Division Multiplexing (OFDM). This MCM improves the operating performance of the system by effectively creating a flat fading environment. With the presence of impulsive noise present in power-line communications, this work introduces a combination of recursive filtering in conjunction with>tht QC-BIBD-LDPC coding to mitigate its effects. To implement the sum-product algorithm (SPA) for decoding Low-Density Parity-Check (LDPC) codes in a power-line communications channel impaired by highly impulsive noise, it is essential to find the right channel probability distribution to optimize the algorithm. However, the computational complexity of the reliability factor is very high. This research proposes the use of a Kalman filter to help cancel out the degradation effect of impulsive noise and consequently approximate the output of the filter as a normal random variable. It is shown that LDPC-coded OFDM system in conjunction with Kalman filtering offers an efficient alternative solution to compensate for the disturbances caused by impulsive noise. Simulation results are provided to demonstrate the performance with the Kalman filter; it is shown that a gain of 15dB is achievable over a system equipped with a channellimiter. The performance evaluation of error-control codes for research and development purposes using software implementation can be a timely and complex process due to computational complex algorithms. Simulating codes to very low BitIBlock error rates is a computationally intense operation, but it is shown that many aspects of simulating LDPC codes are perfect for parallel computation. The speed and accuracy of the QC-LDPC simulations can be greatly increased by utilising the parallel architecture of graphics processing units (GPUs) from conventional "central processing" using CPUs. It is shown that simulating the QC-BIBD-LDPC codes in chapter 3 using Graphics Processing Units simulation speed-ups are achievable when compared to equivalent implementations using central processing units (CPUs). These speedups are related to an increase in simulation complexity, with speedups increasing with an increase in block-size, number of Sum- Product Algorithm (SPA) iterations and number of codewords decoded in parallel.

Design optimality for treatment comparison experiments has been intensively studied by numerous researchers, employing a variety of statistically sound criteria. Their general formulation is based on the idea that optimality functions of the treatment information matrix are invariant to treatment permutation. This implies equal interest in all treatments. In practice, however, there are many experiments where not all treatments are equally important. When selecting a design for such an experiment, it would be better to weight the information gathered on different treatments according to their relative importance and/or interest. This dissertation develops a general theory of weighted design optimality, with special attention to the block design problem. Among others, this study develops and justifies weighted versions of the popular A, E and MV optimality criteria. These are based on the weighted information matrix, also introduced here. Sufficient conditions are derived for block designs to be weighted A, E and MV-optimal for situations where treatments fall into two groups according to two distinct levels of interest, these being important special cases of the "2-weight optimality" problem. Particularly, optimal designs are developed for experiments where one of the treatments is a control. The concept of efficiency balance is also studied in this dissertation. One view of efficiency balance and its generalizations is that unequal treatment replications are chosen to reflect unequal treatment interest. It is revealed that efficiency balance is closely related to the weighted-E approach to design selection. Functions of the canonical efficiency factors may be interpreted as weighted optimality criteria for comparison of designs with the same replication numbers.<br>Ph. D.

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.

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.

Nonparametric tests are used to test hypotheses when the data at hand violate one or more of the assumptions for parametric tests procedures. The test is an ordered alternative (nondecreasing) when there is prior information about the data. It assumes that the underlying distributions are of the same type and therefore differ in location. For example, in dose-response studies, animals are assigned to k groups corresponding to k doses of an experimental drug. The effect of the drug on the animals is likely to increase or decrease with increasing doses. In this case, the ordered alternative is appropriate for the study. In this paper, we propose eight new nonparametric tests useful for testing against nondecreasing order alternatives for a mixed design involving randomized complete block and balanced incomplete block design. These tests involve various modifications of the Jonckheere-Terpstra test (Jonckheere(1952), Terpstra(1954)) and Alvo and Cabilio’s test (1995). Three, four and five treatments were considered with different location parameters under different scenarios. For three and four treatments, 6,12, and 18 blocks were used for the simulation, while 10, 20, and 30 blocks were used for five treatments. Different tests performed best under different block combinations, but overall the standardized last for Alvo outperformed the other test when the number of treatments and number of missing observations per block increases. A simulation study was conducted comparing the powers of the various modification of Jonckheere-Terpstra (Jonckheere(1952), Terpstra(1954)) and Alvo and Cabilio’s (1995) tests under different scenarios. Recommendations are made.

Nonparametric tests have served as robust alternatives to traditional statistical tests with rigid underlying assumptions. If a researcher expects the treatment effects to follow an umbrella alternative, then the test developed in this research will be applicable in the Balanced Incomplete Block Design (Hemmer’s test). It is hypothesized that Hemmer’s test will prove to be more powerful than the Durbin test when the umbrella alternative is true. A mixed design consisting of a Balanced Incomplete Block Design and a Randomized Complete Block Design will also be considered, where two additional test statistics are developed for the umbrella alternative. Monte Carlo simulation studies were conducted using SAS to estimate powers. Various underlying distributions were used with 3, 4, and 5 treatments, and a variety of peaks and mean parameter values. For the mixed design, different ratios of complete to incomplete blocks were considered. Recommendations are given.