Planar functions and commutative semifields

Budaghyan, Lilya; Helleseth, Tor

doi:10.2478/v10127-010-0002-0

Cited by 7 publications

(6 citation statements)

References 22 publications

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“…If f (y ′ )f (y) −1 = u 2 for some u ∈ F * p and y, y ′ ∈ T , then f (y ′ ) = f (uy) which implies y ′ = ±uy and thus y ′ = y. This proves the second half of (2). It follows that the multiset {f (y)F * p |y ∈ T } covers each coset of F * p 2 /F * p at most twice.…”

Section: Preliminariesmentioning

confidence: 60%

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On homogeneous planar functions

Feng

2015

Finite Fields and Their Applications

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Section: Preliminariesmentioning

confidence: 60%

“…Such planar functions are closely related to semifields, and we refer the reader to [2,13] for a most recent survey on this topic, where the reader can also find a list of all known planar functions and some references. For the combinatorial aspects of DO planar functions, we refer to [22,23].…”

Section: Introductionmentioning

confidence: 99%

On homogeneous planar functions

Feng

2015

Finite Fields and Their Applications

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“…[17]. The list of known commutative semifields is short [12,34,49], and Minami and Nakagawa [39] have determined the polynomial forms of certain commutative semifields. It may be of some interest to examine the known commutative semifields one by one to check whether their isotopes will yield planar functions of the form L(X) 2 − wX 2 , but the answer is most probably negative.…”

Section: A Function Approach To C-planes and H-planesmentioning

confidence: 99%

Finite flag-transitive affine planes with a solvable automorphism group

Feng

2017

Journal of Combinatorial Theory, Series A

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In this paper, we consider finite flag-transitive affine planes with a solvable automorphism group. Under a mild number-theoretic condition involving the order and dimension of the plane, the translation complement must contain a linear cyclic subgroup that either is transitive or has two equal-sized orbits on the line at infinity. We develop a new approach to the study of such planes by associating them with planar functions and permutation polynomials in the odd order and even order case respectively. In the odd order case, we characterize the Kantor-Suetake family by using Menichetti's classification of generalized twisted fields and Blokhuis, Lavrauw and Ball's classifcation of rank two commutative semifields. In the even order case, we develop a technique to study permutation polynomials of DO type by quadratic forms and characterize such planes that have dimensions up to four over their kernels. IntroductionLet V be a 2n-dimensional vector space over the finite field F q . A spread S of V is a collection of n-dimensional subspaces that partitions the nonzero vectors in V . The members of S are the components, and V is the ambient space. The kernel is the subring of ΓL(V ) that fixes each component, and it is a finite field containing F q . The dimension of S is the common value of the dimensions of its components over the kernel. The automorphism group Aut(S) is the subgroup of ΓL(V ) that maps components to components. The incidence structure Π(S) with point set V and line set {W + v : W ∈ S, v ∈ V } and incidence being inclusion is a translation plane. The kernel or dimension of Π(S) is that of S respectively. Andre [4] has shown that Aut(S) is the translation complement of the plane Π(S) and each finite translation plane can be obtained from a spread in this way. Two spreads S and S ′ of V are isomorphic if S ′ = {g(W ) : W ∈ S} for some g ∈ ΓL(V ), and isomorphic spreads correspond to isomorphic planes.An affine plane is called flag-transitive if it admits a collineation group which acts transitively on the flags, namely, the incident point-line pairs. Throughout this paper, we will only consider finite planes. Wagner [47] has shown that finite flag-transitive planes are necessarily translation planes, so the plane must have prime power order and can be constructed from a spread S with ambient space V of dimension 2n over F q for some n and q. The affine plane Π(S) constructed from a spread S is flag-transitive if and only if Aut(S) is transitive on the components. Foulser has determined all flag-transitive groups of finite affine planes in [24,25]. The only non-Desarguesian flag-transitive affine planes with nonsolvable collineation groups are the nearfield planes of order 9, the Hering plane of order 27 [23], and the Lüneburg planes of even order [36], cf. [13, 30]. In the solvable case, Foulser has shown that with a finite number of exceptions, which are explicitly described, a solvable flag transitive group of a finite affine plane is a subgroup of a one-dimensional Desarguesian affine plane.Kantor...

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“…The property of being commutative implies that these semifields have applications to perfect nonlinear functions (see e.g. [6] for a survey on planar functions and commutative semifields and for further references). Moreover, R2CS are equivalent to semifield flocks of a quadratic cone in a 3-dimensional projective space.…”

Section: Introductionmentioning

confidence: 99%

Classification of 8-dimensional rank two commutative semifields

Lavrauw

Rodgers

2018

Advances in Geometry

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We classify the rank two commutative semifields which are 8-dimensional over their center Fq. This is done using computational methods utilizing the connection to linear sets in PG(2, q 4 ). We then apply our methods to complete the classification of rank two commutative semifields which are 10-dimensional over F3. The implications of these results are detailed for other geometric structures such as semifield flocks, ovoids of parabolic quadrics, and eggs.

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Planar functions and commutative semifields

Cited by 7 publications

References 22 publications

On homogeneous planar functions

On homogeneous planar functions

Finite flag-transitive affine planes with a solvable automorphism group

Classification of 8-dimensional rank two commutative semifields

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