A k-majority tournament is a directed graph that models a k-majority voting scenario, which is realized by 2k - 1 rankings, called linear orderings, of the vertices in the tournament. Every k-majority voting scenario can be modeled by a tournament, but not every tournament is a model for a k-majority voting scenario. In this thesis we show that all acyclic tournaments can be realized as 2-majority tournaments. Further, we develop methods to realize certain quadratic residue tournaments as k-majority tournaments. Thus, each tournament within these classes of tournaments is a model for a k-majority voting scenario. We also explore important structures specifically pertaining to 2- and 3-majority tournaments and introduce the idea of pseudo-3-majority tournaments and inherited 2-majority tournaments.

In 2010, Prof. Chang and Prof. Lee applied Lagrange interpolation formula to decode a class of binary cyclic codes, but they did not provide an effective way to calculate the Lagrange interpolation formula. In this thesis, we use the least common multiple of polynomials to compute it effectively. Let E be an extension field of degree m over F = F_p and £] be a primitive nth root of unity in E. For a nonzero element r in E, the minimal polynomial of r over F is denoted by m_r(x). Then, let Min (r, F) denote the least common multiple of m_r£]^i(x) for i = 0, 1,..., n-1 and be called the generalized minimal polynomial of over F. For any binary quadratic residue code mentioned in this thesis, the set of all its correctable error patterns can be partitioned into root sets of some generalized minimal polynomials over F. Based on this idea, we can develop an effective method to calculate the Lagrange interpolation formula.

In this paper, we provide a generalization of binary quadratic residue codes to the cases of higher power prime residues over the finite field of the same order, which we will call qth power residue codes. We find generating polynomials for such codes, define a new notion corresponding to the binary concept of an idempotent, and use this to find square root lower bound for the codeword weight of the duals of such codes, which leads to a lower bound on the weight of the codewords themselves. In addition, we construct a family of asymptotically bad qth power residue codes. / text

This work focuses on the issue of acoustic diffusers. The introductory chapter describes the necessary theory of the sound distribution through enclosed space. Acoustic fields are also described. A description of the different diffusion element types and theirs design methods follows. It focuses mainly on design, which uses pseudo-random mathematical sequences. The aim of the work is to produce several types of acoustic diffusors and to verify their diffusion properties by means of measurements. The work uses the AFMG Reflex to simulate the diffusion properties of the proposed elements. Further, the thesis contains a description of the diffusion properties measurement process by the boundary plane method and the process of evaluating the measured data using the Matlab program.

Data security will play a central role in the design of future IT systems. The PC has been a major driver of the digital economy. Recently, there has been a shift towards IT applications realized as embedded systems, because they have proved to be good solutions for many applications, especially those which require data processing in real time. Examples include security for wireless phones, wireless computing, pay-TV, and copy protection schemes for audio/video consumer products and digital cinemas. Most of these embedded applications will be wireless, which makes the communication channel vulnerable. The implementation of cryptographic systems presents several requirements and challenges. For example, the performance of algorithms is often crucial, and guaranteeing security is a formidable challenge. One needs encryption algorithms to run at the transmission rates of the communication links at speeds that are achieved through custom hardware devices. Public-key cryptosystems such as RSA, DSA and DSS have traditionally been used to accomplish secure communication via insecure channels. Elliptic curves are the basis for a relatively new class of public-key schemes. It is predicted that elliptic curve cryptosystems (ECCs) will replace many existing schemes in the near future. The main reason for the attractiveness of ECC is the fact that significantly smaller parameters can be used in ECC than in other competitive system, but with equivalent levels of security. The benefits of having smaller key size include faster computations, and reduction in processing power, storage space and bandwidth. This makes ECC ideal for constrained environments where resources such as power, processing time and memory are limited. The implementation of ECC requires several choices, such as the type of the underlying finite field, algorithms for implementing the finite field arithmetic, the type of the elliptic curve, algorithms for implementing the elliptic curve group operation, and elliptic curve protocols. Many of these selections may have a major impact on overall performance. In this dissertation a finite field from a special class called the Optimal Extension Field (OEF) is chosen as the underlying finite field of implementing ECC. OEFs utilize the fast integer arithmetic available on modern microcontrollers to produce very efficient results without resorting to multiprecision operations or arithmetic using polynomials of large degree. This dissertation discusses the theoretical and implementation issues associated with the development of this finite field in a low end embedded system. It also presents various improvement techniques for OEF arithmetic. The main objectives of this dissertation are to --Implement the functions required to perform the finite field arithmetic operations. -- Implement the functions required to generate an elliptic curve and to embed data on that elliptic curve. -- Implement the functions required to perform the elliptic curve group operation. All of these functions constitute a library that could be used to implement any elliptic curve cryptosystem. In this dissertation this library is implemented in an 8-bit AVR Atmel microcontroller. / Dissertation (MEng (Computer Engineering))--University of Pretoria, 2006. / Electrical, Electronic and Computer Engineering / unrestricted