Guglielmo Lunardon

A New Semifield Flock

We construct the new semifield flock ofPG(3, 243) associated with the Penttila?Williams translation ovoid ofQ(4, 243) and we study the associated generalized quadrangle and its translation dual.

On the flocks of Q+(3,q)

A complete characterization of the flocks of Q+(3, q) is given. As an application, it follows that if q is odd, q≠11, 23, 59, there exist no maximal exterior sets of Q+ (2n−1, q) for n>2.

Generalized Twisted Gabidulin Codes

Abstract Let C be a set of m by n matrices over F q such that the rank of A − B is at least d for all distinct A , B ∈ C . Suppose that m ⩽ n . If # C = q n ( m − d + 1 ) , then C is a maximum rank distance (MRD for short) code. Until 2016, there were only two known constructions of MRD codes for arbitrary 1 d m − 1 . One was found by Delsarte (1978) [8] and Gabidulin (1985) [10] independently, and it was later generalized by Kshevetskiy and Gabidulin (2005) [16] . We often call them (generalized) Gabidulin codes. Another family was recently obtained by Sheekey (2016) [22] , and its elements are called twisted Gabidulin codes. In the same paper, Sheekey also proposed a generalization of the twisted Gabidulin codes. However the equivalence problem for it is not considered, whence it is not clear whether there exist new MRD codes in this generalization. We call the members of this putative larger family generalized twisted Gabidulin codes. In this paper, we first compute the Delsarte duals and adjoint codes of them, then we completely determine the equivalence between different generalized twisted Gabidulin codes. In particular, it can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets of this family.

Translation dual of a semifield

In this paper we obtain a new description of the translation dual of a semifield introduced in [G. Lunardon, Translation ovoids, J. Geom. 76 (2003) 200-215]. Using such a description we are able to prove that a semifield and its translation dual have nuclei of the same order. Combining the Knuth cubical array and the translation dual, we give an alternate description of the chain of twelve semifields in the table of [S. Ball, G.L. Ebert, M. Lavrauw, A geometric construction of finite semifields, J. Algebra 311 (2007) 117-129].

Spreads in H(q) and 1-systems of Q(6,q )

In this paper we prove that the projections along reguli of a translation spread of the classical generalized hexagon H(q) are translation ovoids of Q(4, q). As translation ovoids of Q(4, 2r) are elliptic quadrics, this forces that all translation spreads ofH (2r) are semi-classical. By representing H(q) as a coset geometry, we obtain a characterization of a translation spread in terms of a set of points of PG(3, q) which belong to imaginary chords of a twisted cubic and we construct a new example of a semi-classical spread ofH (2r). Finally, we study the semi-classical locally Hermitian 1-systems ofQ (6, q) which are spreads of Q?(5, q).

Simplectic spreads and finite semifields

It is well known that associated with a translation plane π there is a family of equivalent spreads. In this paper, we prove that if one of these spreads is symplectic and π is finite, then all the associated spreads are symplectic. Also, using the geometric intepretation of the Knuth’s cubical array, we prove that a symplectic semifield spread of dimension n over its left nucleus is associated via a Knuth operation to a commutative semifield of dimension n over its middle nucleus.

Blocking Sets and Derivable Partial Spreads

We prove that a GF(q)-linear Rédei blocking set of size qt + qt−1 + ··· + q + 1 of PG(2,qt) defines a derivable partial spread of PG(2t − 1, q). Using such a relationship, we are able to prove that there are at least two inequivalent Rédei minimal blocking sets of size qt + qt−1 + ··· + q + 1 in PG(2,qt), if t ≥ 4.

Blocking sets and semifields

We find a relationship between semifield spreads of PG(3, q), small Redei minimal blocking sets of PG(2, q2), disjoint from a Baer subline of a Redei line, and translation ovoids of the hermitian surface H(3, q2).

MRD-codes and linear sets

Abstract In this paper we point out the relationship between linear MRD-codes and various geometric objects as linearized polynomials, linear sets of P G ( n − 1 , q n ) , and generalized Segre varieties. We introduce and characterize a class of particular linear MRD-codes called F q n -linear codes with distance n − k + 1 proving that such codes are equivalent to ( n − k + 1 ) -embeddings of a canonical subgeometry.

On non-hyperelliptic flocks

Abstract The prove that a non-hyperelliptic flock of the quadratic cone, the associated translation plane of which is either a semifield or a likeable plane, is isomorphic to one of the known examples.