65 research outputs found
A new semifield of order 2^{10}
No full textIn 2009 N. L. Johnson, G. Marino, O. Polverino and R. Trombetti introduced a family of semifields of order q^{2n}, n > 1 and n odd, with left nucleus F_{q^n} , right and middle nuclei both F_{q^2} and center F_q. Moreover, G. Lunardon exhibited a construction method yielding a theoretical family of order q^{2n} having the same parameters. For n > 3 it is not known if the two families produce isotopic semifields. In this article, the authors prove that for n > 3 any semifield from the theoretical family of semifields introduced by Lunardon is not isotopic to any semifield introduced in the first paper cited above; also, this family is not empty when n > 3, by exhibiting for n = 5 a semifield of order 210, which turns out to be non-isotopic to any other known semifield
On the equivalence issue of a class of 2-dimensional linear maximum rank distance codes
No full textIn [A. Neri, P. Santonastaso, F. Zullo. Extending two families of maximum rank distance codes], the authors extended the family of 2-dimensional Fq2t -linear MRD codes recently found in [G. Longobardi, G. Marino, R. Trombetti, Y. Zhou. A large family of maximum scattered linear sets of PG(1,q^n) and their associated MRD codes]. Also, for t ≥ 5 they determined equivalence classes of the elements in this new family and provided the exact number of inequivalent codes in it. In this article, we complete the study of the equivalence issue removing the restriction t ≥ 5. Moreover, we prove that in the case when t = 4, the linear sets of the projective line PG(1,q8) ensuing from codes in the relevant family, are not equivalent to any one known so far
Semifields of order q^6 with left nucleus F_{q^3} and center F_q
No full textIn [G. Marino, O. Polverino, R. Trombetti, On Fq-linear sets of PG(3,q3) and semifields, J. Combin.
Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families Fi
(i=0,...,5)ofsemifieldsoforderq6withleftnucleusFq3 andcenterFq,accordingtothedifferentgeo-
metric configurations of the associated Fq -linear sets. In this paper we first prove that any semifield of order
q6 with left nucleus Fq3 , right and middle nuclei Fq2 and center Fq is isotopic to a cyclic semifield. Then,
we focus on the family F4 by proving that it can be partitioned into three further non-isotopic families:
F (a) , F (b) , F (c) and we show that any semifield of order q 6 with left nucleus F 3 , right and middle nuclei 444 q
F 2 and center Fq belongs to the family F(c). q4
© 2007 Elsevier Inc. All rights reserved
The isotopism problem of a class of 6-dimensional rank 2 semifields and its solution
No full textIn [U. Dempwolff: \textit{More Translation Planes and Semifields from Dembowski-Ostrom Polynomials}, Designs, Codes, Cryptogr. \textbf{68} (1-3) (2013), 81--103] three classes of rank two presemifields of order , with and odd, were exhibited, leaving as an open problem the isotopy issue. In [M. Lavrauw, G. Marino, O. Polverino, R. Trombetti: \textit{Solution to an isotopism question concerning rank 2 semifields}, J. Comb. Des., \textbf{23} (2015), 60--77], the authors faced with this problem answering the question whether these presemifields are new for . In this paper we complete the study solving the case
Unitals in shift planes of odd order
No full textA finite shift plane can be equivalently defined via abelian relative difference sets as well as planar functions. In this paper, we present a generic way to construct unitals in finite shift planes of odd square order. We investigate various geometric and combinatorial properties of these planes, such as the self-duality, the existence of O’Nan configurations, Wilbrink’s conditions, the designs formed by circles and so on. We also show that our unitals are inequivalent to the unitals derived from unitary polarities in the same shift planes. As designs, our unitals are also not isomorphic to the classical unitals (the Hermitian curves)
Translation ovoids of flock generalized quadrangles
No full textAbstractWe give a characterization of translation ovoids of flock generalized quadrangles. Then we prove that, if q=2e, 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
Nuclei and automorphism groups of generalized twisted Gabidulin codes
No full textGeneralized twisted Gabidulin codes are one of the few known families of maximum rank matrix codes over finite fields. As a subset of m by n matrices, when m=n, the automorphism group of any generalized twisted Gabidulin code has been completely determined by the authors recently. In this paper, we consider the same problem for
On kernels and nuclei of rank metric codes
No full textFor each rank metric code (Formula presented.), we associate a translation structure, the kernel of which is shown to be invariant with respect to the equivalence on rank metric codes. When (Formula presented.) is (Formula presented.)-linear, we also propose and investigate other two invariants called its middle nucleus and right nucleus. When (Formula presented.) is a finite field (Formula presented.) and (Formula presented.) is a maximum rank distance code with minimum distance (Formula presented.) or (Formula presented.), the kernel of the associated translation structure is proved to be (Formula presented.). Furthermore, we also show that the middle nucleus of a linear maximum rank distance code over (Formula presented.) must be a finite field; its right nucleus also has to be a finite field under the condition (Formula presented.). Let (Formula presented.) be the DHO-set associated with a bilinear dimensional dual hyperoval over (Formula presented.). The set (Formula presented.) gives rise to a linear rank metric code, and we show that its kernel and right nucleus are isomorphic to (Formula presented.). Also, its middle nucleus must be a finite field containing (Formula presented.). Moreover, we also consider the kernel and the nuclei of (Formula presented.) where k is a Knuth operation
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