The Generalized Translation Dual of a Semifield

From semifield flocks to the generalized translation dual of a semifield

“…, X n−1 such that dim M = n 2 −n−1. Then M is contained in M(S n,n ) and M(S n,n ) = M ∈T M (see [26]).…”

“…Also, S 2,2 = Q + (3, q) is a hyperbolic quadric and M(S n,n ) = Q + (3, q). Hence if n = 2, the construction described in Theorem 4.2 returns the classical translation dual operation (see [26]).…”

The isotopism problem of a class of 6-dimensional rank 2 semifields and its solution

“…In [29], generalizing results contained in [32], [19] and [24], a family of maximum scattered linear sets, called of pseudoregulus type, is introduced. Precisely, a scattered F q -linear set L U of Λ = P G(2h − 1, q n ) of rank hn (h, n ≥ 2) is of pseoudoregulus type if (i) there exist s 1 , s 2 , .…”

“…, m} is called the F q -pseudoregulus (or simply pseudoregulus) of Λ associated with L U and T 1 and T 2 are the transversal spaces of P L U (or transversal spaces of L U ). Note that by [29,Cor. 3.3], if n > 2 the pseudoregulus P L U associated with L U and its transversal lines are uniquely determined.…”

“…[28], or [22,Section 3]). This operation extends the Knuth orbit and has been recently generalized in [29]. Another recent extension of the Knuth orbit is contained in [26], using the so-called BEL-configurations (see [3], [17], and [18]).…”