On BEL-configurations and finite semifields

The BEL-construction for finite semifields was introduced in Ball et al. (J Algebra 311:117–129, 2007); a geometric method for constructing semifield spreads, using so-called BEL-configurations in $$V(rn,q)$$V(rn,q). In this paper we investigate this construction in greater detail, and determine an explicit multiplication for the semifield associated with a BEL-configuration in $$V(rn,q)$$V(rn,q), extending the results from Ball et al. (2007), where this was obtained only for $$r=n$$r=n. Given a BEL-configuration with associated semifield spread $$\mathcal S$$S, we also show how to find a BEL-configuration corresponding to the dual spread $$\mathcal {S}^{\epsilon }$$Sϵ. Furthermore, we study the effect of polarities in $$V(rn,q)$$V(rn,q) on BEL-configurations, leading to a characterisation of BEL-configurations associated to symplectic semifields. We give precise conditions for when two BEL-configurations in $$V(n^2,q)$$V(n2,q) define isotopic semifields. We define operations which preserve the BEL property, and show how non-isotopic semifields can be equivalent under this operation. We also define an extension of the “switching” operation on BEL-configurations in $$V(2n,q)$$V(2n,q) introduced in Ball et al. (2007), which, together with the transpose operation, leads to a group of order $$8$$8 acting on BEL-configurations.

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

[2]  John Sheekey,et al.  On embeddings of minimum dimension of PG(n,q)×PG(n,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{PG}}(n,q , 2013, Designs, Codes and Cryptography.

[3]  Giuseppe Marino,et al.  Semifields of order q6 with left nucleus Fq3 and center Fq , 2008, Finite Fields Their Appl..

[4]  M. Lavrauw.  The two sets of three semifields associated with a semifield flock , 2005 .

[5]  Giuseppe Marino,et al.  Translation dual of a semifield , 2008, J. Comb. Theory, Ser. A.

[6]  Michel Lavrauw,et al.  Finite semifields and nonsingular tensors , 2013, Des. Codes Cryptogr..

[7]  Norman L. Johnson,et al.  The Collineation Groups of Generalized Twisted Field Planes , 1999 .

[8]  Michel Lavrauw,et al.  A geometric construction of finite semifields , 2007 .

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

[10]  Giuseppe Marino,et al.  On the multiplication of some semifields of order q6 , 2009, Finite Fields Their Appl..

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

[12]  Michel Lavrauw,et al.  Field reduction and linear sets in finite geometry , 2013, 1310.8522.

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

[14]  Simeon Ball,et al.  The six semifield planes associated with a semifield flock , 2004 .

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

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

[17]  A. Barlotti,et al.  Finite Sperner spaces constructed from projective and affine spaces , 1974 .