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Finite fields and Galois geometries

Book Contribution - Book Chapter Conference Contribution

In 1954 Segre proved the following celebrated theorem : In PG(2, q), with q odd, every oval is a nonsingular conic. Crucial for the proof is Segre's Lemma of Tangents, where a strong result is deduced from the simple fact that the product of the nonzero elements of GF(q) is -1. Relying on this Lemma of Tangents he was able to prove excellent theorems on certain point sets in PG(2,q). To this end he also generalized the classical theorem of Menelaus to an arbitrary collection of lines in the plane PC (2, q), no three of which are concurrent. As a corollary of these theorems good results on linear MDS codes were obtained. Here we review generalizations of the Lemma of Tangents, generalizations of Segre's generalization of the theorem of Menelaus, and applications to Hermitian curves, semiovals, circle geometries and linear MDS codes. Finally we report on recent research about generalized ovals: the elements of a generalized oval are subspaces of a projective space. To do this an appropriate 'Lemma of Tangents' type theorem is proved.
Book: Contemporary Mathematics Series
Volume: 461
Pages: 251 - 265
Publication year:2008