t-blocking sets and semiovals in Witt designs

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Bao, Xiaomin
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The purpose of this thesis is to characterize all 't'-blocking sets (see definition 3.1) and all semiovals (see definition 4.2) in the Witt designs up to the frequency vectors. In chapter 2 we obtain some results which correct some mistakes in [7, 8]. We also improve the results in [9]. In chapter 3 we thoroughly determine the structure of a Fano set--a fundamental structure used to characterize blocking sets, 't'-blocking sets and semiovals in ' S'(3,6,22), and obtain a construction method for a Fano set. Then we characterize the 't'-blocking sets in Witt designs. We determine the possible size of a 't'-blocking set, classifying them by their frequency vectors. We also analyze the Witt designs and obtain a number of new results. In chapter 4, we characterize all semiovals in the Witt designs. We prove that there are only three types of semiovals in 'S'(3,6,22), one of them has size nine, the other two have size ten, one of them is also a 't'-blocking set; 'S'(5,6,12) and ' S'(4,7,23) each have only one type of semioval, which are also ' t'-blocking sets; but both 'S'(4, 5, 11) and ' S'(5, 8, 24) have no semioval at all. (Abstract shortened by UMI.)