Steiner 3-Designs

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Steiner 2-designs are analogous to ordinary planes, (Euclidean or projective plane), except that the number of points and the number of lines is finite. The defining properties are (i) any two points is contained in exactly one line and (ii) all lines (viewed as subsets of points) contain the same number of points. In statistical terminology, points are same as treatments and lines are called blocks and the 2-design is called an incomplete block design. Incomplete block designs (or 2-designs) are needed to construct optimum experimental plans to determine the best treatment. Therefore it is important to determine for what parameters v, (the total number of points) and k (the number of points), a Steiner 2-design can be constructed. After decades of research culminating in the work of R. M. Wilson, the existence and construction problem for Steiner 2-designs is solved for at least large number of points. Steiner 3-designs are analogous to circle geometry in which any 3-points is contained in a unique circle, where the numbers of points and circles (blocks) are finite. Steiner 3-designs are fundamental mathematical objects which have application in statistical experimental plans, mathematical communication theory (coding theory), cryptography, computer networking and many branches of mathematics. The problem of determining parameters, v (the total number of points) and k, the circle (block) size for which a Steiner 3-design exists and the corresponding construction problem are very challenging problems. The PI and his collaborators already made significant progress on this problem. The PI now proposes to solve the existence problem for large v. Historically, continuous mathematics, i.e. calculus, differential equations, etc. played a fundamental role in the formulation of physical phenomena and their idealized solutions. These kinds of mathematical approaches solved many practical problems of engineering like building of bridges, aircraft and space ships, electrical networks and also provided the foundation of theoretical physics. With the advent of computers, methods of 'discrete mathematics' where one studies a space of finitely many points are playing increasingly important roles in diverse areas like planning optimum clinical trials (experimental designs), scheduling, networks, cryptography, etc. A Steiner 2- design is like Euclidean geometry except that the number of points and lines are finite. Steiner 3-designs studies 3-dimensional configurations in finite geometry. Steiner designs have important applications in optimum planning of trials, computer networks, communication in the presence of noise, and cryptography. This project will concentrate on mathematical methods for construction of Steiner designs.