Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries

In this paper we construct infinite families of non-linear maximum rank distance codes by using the setting of bilinear forms of a finite vector space. We also give a geometric description of such codes by using the cyclic model for the field reduction of finite geometries and we show that these families contain the non-linear maximum rank distance codes recently provided by Cossidente, Marino and Pavese.Comment: submitted; 22 page

Lo scalo artistico del disagio adolescenziale. L’esperienza bolognese della STAV

Nata nel 2010 dall’incontro tra psicoanalisi e arte, la Scuola di Teatro e Arti Video-grafiche (STAV) si occupa della realizzazione di laboratori musicali e artistici. È un luogo in cui l’arte e gli artisti intendono incontrare giovani adolescenti abitati da dolori e sofferenze psicosociali. Arte come “linguaggio” del trattamento. "The 'scalo artistico' of teenage distress. The experience of STAV in Bologna". Born in 2010 from the encounter between psychoanalysis and art, the Theatre and Videographic Arts School provides musical and artistic workshops. It is a place where teenagers with psychosocial distress can seek treatment through the lan- guage/expression of art

(B)-Geometries and flocks of hyperbolic quadrics

AbstractWe give a characteristic-free proof of the classification theorem for flocks of hyperbolic quadrics of PG(3,q)

Embedding of orthogonal Buekenhout-Metz unitals in the Desarguesian plane of order q^2

A unital, that is a 2-(q^3 + 1, q + 1, 1) block-design, is embedded in a projective plane π of order q^2 if its points are points of π and its blocks are subsets of lines of π, the point-block incidences being the same as in π. Regarding unitals U which are isomorphic, as a block-design, to the classical unital, T. Szonyi and the authors recently proved that the natural embedding is the unique embedding of U into the Desarguesian plane of order q^2. In this paper we extend this uniqueness result to all unitals which are isomorphic, as block-designs, to orthogonal Buekenhout-Metz unitals

On the incidence map of incidence structures

By using elementary linear algebra methods we exploit properties of the incidence map of certain incidence structures with finite block sizes. We give new and simple proofs of theorems of Kantor and Lehrer, and their infinitary version. Similar results are obtained also for diagrams geometries. By mean of an extension of Block’s Lemma on the number of orbits of an automorphism group of an incidence structure, we give informations on the number of orbits of: a permutation group (of possible infinite degree) on subsets of finite size; a collineation group of a projective and affine space (of possible infinite dimension) over a finite field on subspaces of finite dimension; a group of isometries of a classical polar space (of possible infinite rank) over a finite field on totally isotropic subspaces (or singular in case of orthogonal spaces) of finite dimension. Furthermore, when the structure is finite and the associated incidence matrix has full rank, we give an alternative proof of a result of Camina and Siemons. We then deduce that certain families of incidence structures have no sharply transitive sets of automorphisms acting on blocks

Bol quasifields

In the context of configurational characterisations of symmetric projective planes, a new proof of a theorem of Kallaher and Ostrom characterising planes of even order of Lenz-Barlotti type IV.a.2 via Bol conditions is given. In contrast to their proof,we need neither the Feit-Thompson theorem on solvability of groups of odd order, nor Bender’s strongly embedded subgroup theorem, depending rather on Glauberman’s Z*-theorem

Bol quasifields

In the context of configurational characterisations of symmetric projective planes, a new proof of a theorem of Kallaher and Ostrom characterising planes of even order of Lenz-Barlotti type IV.a.2 via Bol conditions is given. In contrast to their proof,we need neither the Feit-Thompson theorem on solvability of groups of odd order, nor Bender’s strongly embedded subgroup theorem, depending rather on Glauberman’s Z*-theorem

Pseudo-ovals of elliptic quadrics as Delsarte designs of association schemes

A pseudo-oval of a finite projective space over a finite field of odd order q is a configuration of equidimensional subspaces that is essentially equivalent to a translation generalised quadrangle of order (q^n,q^n) and a Laguerre plane of order (for some n). In setting out a programme to construct new generalised quadrangles, Shult and Thas asked whether there are pseudo-ovals consisting only of lines of an elliptic quadric Q^-(5,q) , non-equivalent to the classical example, a so-called pseudo-conic. To date, every known pseudo-oval of lines of Q^-(5,q) is projectively equivalent to a pseudo-conic. Thas characterised pseudo-conics as pseudo-ovals satisfying the perspective property, and this paper is on characterisations of pseudo-conics from an algebraic combinatorial point of view. In particular, we show that pseudo-ovals in Q^-(5,q) and pseudo-conics can be characterised as certain Delsarte designs of an interesting five-class association scheme. These association schemes are introduced and explored, and we provide a complete theory of how pseudo-ovals of lines of Q^-(5,q) can be analysed from this viewpoint

Classification of flocks of the quadratic cone in PG(3,64)

Flocks are an important topic in the field of finite geometry, with many relations with other objects of interest. This paper is a contribution to the difficult problem of classifying flocks up to projective equivalence. We complete the classification of flocks of the quadratic cone in PG(3,q) for q ≤ 71, by showing by computer that there are exactly three flocks of the quadratic cone in PG(3,64), up to equivalence. The three flocks had previously been discovered, and they are the linear flock, the Subiaco flock and the Adelaide flock. The classification proceeds via the connection between flocks and herds of ovals in PG(2,q), q even, and uses the prior classification of hyperovals in PG(2, 64)