Topics in Algebraic Design Theory

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The project addresses several problems of central importance in the areas of designs, codes and association schemes. The emphasis will be on the use of algebraic and number theoretic methods in the above areas. Topics of the proposed research include Smith normal forms of subspace-inclusion matrices (q-analogues of the subset-inclusion matrices); connections between difference sets and maximal arcs in projective spaces; strongly regular graphs, two-weight codes and association schemes. Number theoretic tools, such as Gauss and Jacobi sums, have been used with success by the PI and others in all of the afforementioned topics. It is expected that deeper p-adic number theory and representation theory of the general linear group will be necessary in the further investigations of these topics. Combinatorial designs first arose in recreational mathematics problems, such as Kirkman's 15 schoolgirls problem, and later in the design of statistical experiments. They found many applications in coding theory, finite geometry, cryptography, computer science, and electrical engineering. Designs and codes are intimately related. Many efficient error-correcting codes are constructed from designs. These codes are used nowadays in our daily life, for example, in CD players, high-speed modems, and cellular phones. This proposal investigates several problems in the interface of designs and codes, and gives summer support for a graduate student who will study and work in the areas of design theory and coding theory related to the above mentioned applications.