We give a characterization of translation ovoids of flock generalized quadrangles. Then we prove that, if q = 2 e , each elation generalized quadrangle defined by a flock has a large class of translation ovoids which arise from semifields of dimension two over their left nucleus. Finally, we give some examples when q is odd.
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… (More)
Based on the twisted Gabidulin codes obtained recently by Sheekey, we construct a new family of maximal rank distance codes as a set of q-polynomials over F q n , which includes the generalized Gabidulin codes and the twisted Gabidulin codes. Their Delsarte duals and adjoint codes are investigated. We also obtain necessary and sufficient conditions for the… (More)
All semifields 6–dimensional over the center F q , with one nucleus of order q 3 and at least one of the remaining nuclei of order q 2 , are classified. The main result represents a further non–trivial step towards the complete classification of rank 2 semifields 6–dimensional over their center.
For each rank metric code C ⊆ K m×n , we associate a translation structure, the kernel of which is showed to be invariant with respect to the equivalence on rank metric codes. When C is K-linear, we also propose and investigate other two invariants called its middle nucleus and right nucleus. When K is a finite field Fq and C is a maximum rank distance code… (More)
There are lovely connections between certain characteristic 2 semi-fields and their associated translation planes and orthogonal spreads on the one hand, and Z 4 –linear Kerdock and Preparata codes on the other. These inter– relationships lead to the construction of large numbers of objects of each type. In the geometric context we construct and study large… (More)