Grimaldi, Giovanni Giuseppe (2024) The dark side of finite geometry: σ-quadrics, m-ovoids and non-linear MRD codes. [Tesi di dottorato]

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Abstract

The aim of this PhD Thesis is to collect the results which I obtained during my PhD course at University of Naples Federico II under the supervision of Prof. Nicola Durante. More precisely, in the first chapter we recall some definitions and notions regarding projective spaces, polarities and polar spaces, and we also give an overview on sesquilinear forms on vector spaces. Finally, we introduce the concept of m-ovoid of a finite classical polar space and describe the link between the ovoids of the Klein quadric and the line spreads of a projective space PG(3, q). In the second chapter, we deal with the theory of maximum rank distance codes (or for short MRD-codes). We present the setting of σ-linearized polynomials, the state of art of the known families of MRD codes and we show how to construct families of RD-codes from exterior sets. In Chapters 3,4,5, the so called σ-quadrics are described. In particular, we give a complete classification of correlations, associated to non-reflexive sesquilinear forms, of a projective line PG(1, qn ) and a projective plane PG(2, qn ) in the degenerate and the non-degenerate case. Also, the classification of correlations, associated to degenerate non-reflexive sesquilinear forms, of a projective space PG(3, qn ) is given. Then, we study correlations associated to degenerate non-reflexive sesquilinear forms, of the projective spaces PG(4, qn ) and PG(5, qn ), respectively. Moreover, we show how σ-quadrics are related to ovoids of polar spaces. In Chapters 6 and 7, we deal the classification of "low-degree" ovoids of the hyperbolic quadric Q+(5, q) and the parabolic quadric Q(6, q). To every ovoid of Q+(5, q) two bivariate polynomials f1(x, y) and f2(x, y) can be associated. We classify ovoids of Q+(5, q) such that f1(x, y) = y + g(x) and max{deg(f1), deg(f2)} is "low" compared with q. It is well known that to any ovoid of Q(6, q) two polynomials f1(x, y, z), f2(x, y, z) can be associated. We classify ovoids of Q(6, q) with max{deg(f1), deg(f2)} is "low" compared with q. In Chapter 8, we provide a construction of (q + 1)-ovoids of the hyperbolic quadric Q+(7, q), q an odd prime power, by "glueing" (q + 1)/2-ovoids of the elliptic quadric Q−(5, q). Secondly, we also construct m-ovoids for m ∈ {2, 4, 6, 8, 10} in Q+(7, 3). Therefore we first investigate how to construct spreads of PG(3, q) that have as many secants to an elliptic quadric as possible. In Chapter 9, we give a geometric construction of a new family of non-linear MRD codes. These codes are obtained from a cone of a projective space whose vertex is a proper subspace and whose base is a special subset of a subspace skew with the vertex. The base of such a cone is linked to the non-linear MRD codes constructed by Cossidente et al., Durante and Siciliano and Donati and Durante. These latter can be obtained in turn by puncturing codes in relevant family. Finally, we show that this class of codes is not equivalent to the one constructed by Otal and Özbudak.

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