Maximum scattered linear sets of pseudoregulus type and the Segre variety $\mathcal{S}_{n,n}$

In this paper we study a family of scattered $\mathbb{F}_{q}$-linear sets of rank tn of the projective space PG(2n−1,qt) (n≥1, t≥3), called of pseudoregulus type, generalizing results contained in Lavrauw and van de Voorde, Des. Codes Crypt. 20(1) (2013) and in Marino et al. J. Combin. Theory, Ser. A 114:769–788 (2007). As an application, we characterize, in terms of the associated linear sets, some classical families of semifields: the Generalized Twisted Fields and the 2-dimensional Knuth semifields.

[1]  Giuseppe Marino,et al.  F q -pseudoreguli of PG ( 3 , q 3 ) and scattered semifields of order q 6 , 2011 .

[2]  Guglielmo Lunardon,et al.  Translation ovoids of orthogonal polar spaces , 2004 .

[3]  Giuseppe Marino,et al.  On Fq-linear sets of PG(3, q3) and semifields , 2007, J. Comb. Theory, Ser. A.

[4]  G. Lunardon.  Translation ovoids , 2003 .

[5]  P. Dembowski.  Finite geometries , 1997 .

[6]  J. W. Freeman.  Reguli and pseudo-reguli in PG(3, s2) , 1980 .

[7]  M. Lavrauw,et al.  Finite Semifields and Galois Geometry ∗ , 2011 .

[8]  Guglielmo Lunardon,et al.  Desarguesian spreads , 2011 .

[9]  L. Storme,et al.  Current Research Topics on Galois Geometry , 2011 .

[10]  Michel Lavrauw,et al.  On the isotopism classes of finite semifields , 2008, Finite Fields Their Appl..

[11]  Rudolf Lide,et al.  Finite fields , 1983 .

[12]  J. Thas,et al.  General Galois geometries , 1992 .

[13]  Beniamino Segre,et al.  Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane , 1964 .

[14]  Olga Polverino,et al.  Semifield planes of order q4 with kernel Fq2 and center Fq , 2006, Eur. J. Comb..

[15]  Michel Lavrauw,et al.  Scattered Spaces with Respect to a Spread in PG(n,q) , 2000 .

[16]  A. Adrian Albert,et al.  Generalized twisted fields. , 1961 .

[17]  Nicola Durante,et al.  Scattered linear sets generated by collineations between pencils of lines , 2014 .

[18]  Michel Lavrauw,et al.  Scattered Linear Sets and Pseudoreguli , 2013, Electron. J. Comb..

[19]  Additive decompositions induced by multiplicative characters over finite fields , 2011 .

[20]  M. Lavrauw.  Finite semifields with a large nucleus and higher secant varieties to Segre varieties , 2011 .

[21]  D. Knuth.  Finite semifields and projective planes , 1965 .

[22]  Michel Lavrauw,et al.  On linear sets on a projective line , 2010, Des. Codes Cryptogr..

[23]  Olga Polverino,et al.  Linear sets in finite projective spaces , 2010, Discret. Math..

[24]  Giuseppe Marino,et al.  Fq-pseudoreguli of PG(3, q3) and scattered semifields of order q6 , 2011, Finite Fields Their Appl..

[25]  Guglielmo Lunardon,et al.  Blocking Sets and Derivable Partial Spreads , 2001 .

[26]  Giuseppe Marino,et al.  On the nuclei of a finite semifield , 2011, 1111.3217.

[27]  Joe W. Harris,et al.  Algebraic Geometry: A First Course , 1995 .