A new family of semifields with 2 parameters. (English) Zbl 1296.12007

In this paper, the authors obtain a new family of commutative semifields defined by two parameters by altering the construction proposed by S. D. Cohen and M. J. Ganley [J. Algebra 75, 373–385 (1982; Zbl 0499.12021)].
Basically, they replace some of the field multiplications which appear in the construction of Cohen-Ganley by the multiplication of Albert’s twisted fields and determine the necessary conditions to have a semifield as a result. The left and the middle nucleus of the resulting semifields are obtained and the isotopism problem is investigated. It is shown that for a particular choice of one of the parameters, the resulting family belongs to the family discovered by L. Budaghyan and T. Helleseth [Sequences and their applications – SETA 2008, Lect. Notes Comput. Sci. 5203, 403–414 (2008; Zbl 1177.94137)] but for the rest of the parameters, the constructed family contains semifields which are not isotopic to any of the known families.
In the last part of the paper, it is shown that one can use this family of semifields to construct APN functions and by choosing the parameters of the semifield appropriately a new APN function on \(\mathbb{F}_{2^8}\) is obtained.


12K10 Semifields
51A35 Non-Desarguesian affine and projective planes
51A40 Translation planes and spreads in linear incidence geometry


Full Text: DOI arXiv


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