Rocco Trombetti

Semifield planes of order q 4 with kernel F q 2 and center F q

A classification of semifield planes of order q4 with kernel Fq2 and center Fq is given. For q an odd prime, this proves the conjecture stated in [M. Cordero, R. Figueroa, On the semifield planes of order 54 and dimension 2 over the kernel, Note Mat. (in press)]. Also, we extend the classification of semifield planes lifted from Desarguesian planes of order q2, q odd, obtained in [M. Cordero, R. Figueroa, On some new classes of semifield planes, Osaka J. Math. 30 (1993) 171-178], to the even characteristic case.

Infinite families of new semifields

We construct six new infinite families of finite semifields, all of which are two-dimensional over their left nuclei. We give constructions for both even and odd characteristics when the left nucleus has odd dimension over the center. The characteristic is odd in the one family in which the left nucleus has even dimension over the center. Spread sets of linear maps are used in all the constructions.

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).

Fq-pseudoreguli of PG(3,q3) and scattered semifields of order q6

In this paper, we study rank two semifields of order q^6 that are of scattered type. The known examples of such semifields are some Knuth semifields, some Generalized Twisted Fields and the semifields recently constructed in Marino et al. (in press) [12] for q=1(mod3). Here, we construct new infinite families of rank two scattered semifields for any q odd prime power, with q=1(mod3); for any q=2^2^h, such that h=1(mod3) and for any q=3^h with h@?0(mod3). Both the construction and the proof that these semifields are new, rely on the structure of the linear set and the so-called pseudoregulus associated to these semifields.

On the multiplication of some semifields of order q 6

The semifields of order q^6 which are two-dimensional over their left nucleus and six-dimensional over their center have been geometrically partitioned into six classes by using the associated linear sets in PG(3,q^3). One of these classes has been partitioned further into three subclasses. In this paper the generic multiplication is determined for each of these three subclasses, and several examples of new semifields are constructed that belong to these subclasses. For two of the subclasses, no examples were previously known.

On kernels and nuclei of rank metric codes

For each rank metric code

Spreads of PG(3, q) and ovoids of polar spaces

\mathcal {C}\subseteq \mathbb {K}^{m\times n}