Vol. 6, No. 1, 2008

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On semifields of type $(q^{2n},q^{n},q^2,q^2,q)$, $n$ odd

Giuseppe Marino, Olga Polverino and Rocco Trombetti

Vol. 6 (2008), No. 1, 271–289
DOI: 10.2140/iig.2008.6.271
Abstract

A semifield of type (q2n,qn,q2,q2,q) (with n > 1) is a finite semifield of order q2n (q a prime power) with left nucleus of order qn, right and middle nuclei both of order q2 and center of order q. Semifields of type (q6,q3,q2,q2,q) have been completely classified by the authors and N. L. Johnson. In this paper we determine, up to isotopy, the form of any semifield of type (q2n,qn,q2,q2,q) when n is an odd integer, proving that there exist n1 2 non isotopic potential families of semifields of this type. Also, we provide, with the aid of the computer, new examples of semifields of type (q14,q7,q2,q2,q), when q = 2.

Keywords
semifield, isotopy, linear set
Mathematical Subject Classification 2000
Primary: 12K10, 51A40, 51E20
Milestones
Received: 28 February 2008
Accepted: 25 March 2008
Authors
Giuseppe Marino
Olga Polverino
Rocco Trombetti