Abstract
We construct and describe the basic properties of a family of semifields in characteristic 2. The construction relies on the properties of projective polynomials over finite fields. We start by associating non-associative products to each such polynomial. The resulting presemifields form the degenerate case of our family. They are isotopic to the Knuth semifields which are quadratic over left and right nuclei. The non-degenerate members of our family display a very different behavior. Their left and right nuclei agree with the center, the middle nucleus is quadratic over the center. None of those semifields is isotopic or Knuth equivalent to a commutative semifield. As a by-product we obtain the complete taxonomy of the characteristic 2 semifields which are quadratic over the middle nucleus, bi-quadratic over the left and right nuclei and not isotopic to twisted fields. This includes determining when two such semifields are isotopic and the order of the autotopism group.
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The research of D. Bartoli, M. Giulietti, S. Marcugini, and F. Pambianco was supported in part by Ministry for Education, University and Research of Italy (MIUR) (Project PRIN 2012 ”Geometrie di Galois e strutture di incidenza”) and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM). The work of D. Bartoli was supported also by the European Community under a Marie-Curie Intra-European Fellowship (FACE Project: Number 626511). J. Bierbrauer’s research was supported in part by GNSAGA - INdAM.
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Bartoli, D., Bierbrauer, J., Kyureghyan, G. et al. A family of semifields in characteristic 2. J Algebr Comb 45, 455–473 (2017). https://doi.org/10.1007/s10801-016-0713-7
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DOI: https://doi.org/10.1007/s10801-016-0713-7