## Commutative semifields from projection mappings.(English)Zbl 1241.12004

Let $$q$$ be a power of the odd prime $$p$$, $$m$$ an odd natural number, and let $$F$$ be the field with $$q^{2m}$$ elements. Sitting inside $$F$$ are the fields $$F_q$$, $$K$$, $$L$$, with $$q$$, $$q^2$$, $$q^m$$ elements, respectively.
For any map $$f: F\to F$$ define the polarization $$x\ast y:=(f(x+y)-f(x)-f(y))/2$$. Then $$f$$ is called planar if the map $$x\mapsto x\ast a$$ is bijective for each $$a\in F\setminus\{0\}$$. Furthermore, $$f$$ is called quadratic if a polynomial representation $$f(x)=\sum a_ix^i$$ has the property $$a_i\neq 0\Longrightarrow i$$ is a sum of two powers of $$p$$. It is an easy exercise to show that the polarization of a quadratic map is symmetric bilinear over the prime field of $$F$$. Thus the polarization of a quadratic planar map is a commutative presemifield multiplication on $$F$$.
The author constructs a quadratic planar map $$f$$ based on the trace $$F\to L$$ in such a way that $$F_q$$ is contained in the left nucleus and $$K$$ in the middle nucleus of the semifield $$(F,+,\ast)$$. Moreover, we have $$u\ast x=ux$$ for all $$u\in K$$.
This way the author generalizes a construction of G. Lunardon et al. [“Symplectic spreads and quadric veroneseans” (manuscript)].
Modifying his own approach the author also gives a new way to generalize and simplify the construction of a family of commutative semifields due to L. Budaghyan and T. Helleseth [“New commutative semifields defined by new PN multinomials”, Cryptogr. Commun. 3, 1–16 (2011; Zbl 1291.12006); “New perfect nonlinear multinomials over $$\mathbb F_{p^{2k}}$$ for any odd prime $$p$$”, Lect. Notes Comput. Sci. 5203, 403–414 (2008; Zbl 1177.94137)].

### MSC:

 12K10 Semifields 51E15 Finite affine and projective planes (geometric aspects) 51A40 Translation planes and spreads in linear incidence geometry

### Citations:

Zbl 1177.94137; Zbl 1291.12006
Full Text:

### References:

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